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MODULE 1 - THE SET OF REAL NUMBERS
OVERVIEW
The first part of this module serves as a review of your basic knowledge about the topics that you have learned in high school. These are the following: Integer exponents, Operations of Polynomials, Special Products, and Factoring Polynomials. It is required to have a good and strong foundation of the aforementioned topics so that you can be able to understand the succeeding lessons with ease. In the second part, you will be able to review your prior knowledge about radicals. You willstudy how an expression with a rational exponent can be expressed into radical form. Also, you will deal with simplifying radical expressions and perform operations on radicals.
The first part of this module serves as a review of your basic knowledge about the topics that you have learned in high school. These are the following: Integer exponents, Operations of Polynomials, Special Products, and Factoring Polynomials. It is required to have a good and strong foundation of the aforementioned topics so that you can be able to understand the succeeding lessons with ease. In the second part, you will be able to review your prior knowledge about radicals. You willstudy how an expression with a rational exponent can be expressed into radical form. Also, you will deal with simplifying radical expressions and perform operations on radicals.
LEARNING OBJECTIVES:
At the end of this module, you should be able to:1. Simplify exponential expressions.2. Identify the properties of exponents used in simplifying exponential expressions.3. Perform operations of polynomials.4. Factor different types of polynomials.5. Convert expressions with rational exponents into radical expression and vice versa.6. Simplify radical expressions.7. Perform operations on radical expressions.
At the end of this module, you should be able to:1. Simplify exponential expressions.2. Identify the properties of exponents used in simplifying exponential expressions.3. Perform operations of polynomials.4. Factor different types of polynomials.5. Convert expressions with rational exponents into radical expression and vice versa.6. Simplify radical expressions.7. Perform operations on radical expressions.
INTEGER EXPONENTS
As stated in the overview, the first subtopic for this module is about the integer exponents. When we say integer exponent, it is simply the exponents that are integers. Though it is the basic of algebra, this subtopic was a bit challenging for me than the other three subtopics since there are a lot of exponential properties to be considered when solving. First, the zero power property which states that any number (except zero) raised to zero is always equal to 1. Second, the negative power property which states that any number raised to a negative power is equal to its inverse or simply reciprocal. Third, the product of a power property which states that when multiplying powers with the same base, simply copy the base and add the exponents. Fourth, the power of a power property which states that if the given power is raised by another exponent, then the result is the base raised to the product of the exponents. Fifth, the power of a product property which states that if the product of the bases is raised to an exponent then the factor of each product is raised to that exponent. Sixth, the quotient of a power property which states that the quotient of two powers with the same base is equal to the base raised to the difference of their exponents. Seventh and last, the power of a quotient property which states that the quotient of two non-zero real numbers raised to an exponent is equal to the quotient of each base raised to that exponent.
As stated in the overview, the first subtopic for this module is about the integer exponents. When we say integer exponent, it is simply the exponents that are integers. Though it is the basic of algebra, this subtopic was a bit challenging for me than the other three subtopics since there are a lot of exponential properties to be considered when solving. First, the zero power property which states that any number (except zero) raised to zero is always equal to 1. Second, the negative power property which states that any number raised to a negative power is equal to its inverse or simply reciprocal. Third, the product of a power property which states that when multiplying powers with the same base, simply copy the base and add the exponents. Fourth, the power of a power property which states that if the given power is raised by another exponent, then the result is the base raised to the product of the exponents. Fifth, the power of a product property which states that if the product of the bases is raised to an exponent then the factor of each product is raised to that exponent. Sixth, the quotient of a power property which states that the quotient of two powers with the same base is equal to the base raised to the difference of their exponents. Seventh and last, the power of a quotient property which states that the quotient of two non-zero real numbers raised to an exponent is equal to the quotient of each base raised to that exponent.
The above problem, taken from my problem set # 1, would be a great example of integer exponents since almost all the exponential properties could be applied. At first glance, one could be problematic seeing the quite long problem, but by doing step-by-step process, one could get the final answer. On the right was my solution. Firstly, I applied the power of a power property in both numerators and denominators of the two factors where it is applicable in which I just calculated the products of the exponents (second row). Secondly, I dealt with the product of a power property in which I combined the ones with the same bases by simply copying the base and adding the exponents (third row). Thirdly, I applied three properties at the same time, the negative power property, the quotient of a power property, and the product of a power property (fourth row). Fourthly, I simply combined the two factors by multiplying the numerator to numerator and denominator to denominator whilst applying the product of a power property (fifth row), Lastly, I just applied the quotient of a power property by simply subtracting the exponents of same bases, and also the negative power property by getting the reciprocal of the said base (last row).
OPERATIONS OF POLYNOMIALS
The second subtopic is about the operations of polynomials. A polynomial is an algebraic expression which is often a sum of terms that contain coefficients, variables, and exponents. The four basic mathematical operations that will be used are addition, subtraction, multiplication, and division. Addition of polynomials is simply combining like terms by adding the coefficients. Subtraction of polynomials is the same way as adding polynomials except that you need to change the sign of each term into “+“ into “-“or vice versa before adding their coefficients. Meanwhile, multiplication of polynomials is multiplying each term of the polynomial to each term of the other polynomial by applying the distributive property, then, combining like terms if possible to simplify the polynomial. On the other hand, division of polynomials includes dividing a polynomial by a monomial and dividing a polynomial by another polynomial. When dividing a polynomial by a monomial, you must divide each term of the polynomial with the monomial. When dividing a polynomial by another polynomial, there are steps to follow. See the example below.
The second subtopic is about the operations of polynomials. A polynomial is an algebraic expression which is often a sum of terms that contain coefficients, variables, and exponents. The four basic mathematical operations that will be used are addition, subtraction, multiplication, and division. Addition of polynomials is simply combining like terms by adding the coefficients. Subtraction of polynomials is the same way as adding polynomials except that you need to change the sign of each term into “+“ into “-“or vice versa before adding their coefficients. Meanwhile, multiplication of polynomials is multiplying each term of the polynomial to each term of the other polynomial by applying the distributive property, then, combining like terms if possible to simplify the polynomial. On the other hand, division of polynomials includes dividing a polynomial by a monomial and dividing a polynomial by another polynomial. When dividing a polynomial by a monomial, you must divide each term of the polynomial with the monomial. When dividing a polynomial by another polynomial, there are steps to follow. See the example below.
To divide the problem above, I just followed the five steps of dividing polynomial by another polynomial. Here are the steps:1. Arrange the terms of the dividend P(x) and the divisor D(x) in descending order. If there is/are missing term/s, write it as 0xn term to fill in the missing term. For instance, if there is a missing x3 term, then write it as 0x3.2. Divide the first term of the dividend P(x) to the first term of the divisor D(x) to get the first term of the quotient.3. Multiply the first term of the quotient to the divisor then subtract the product from the dividend. Bring down the next term.4. Repeats steps 2 and 3 to get the next term of the quotient. Continue doing these steps until there are no more terms to bring down.5. Write the final answer. In case there is a remainder R(x), it should be written as: Q(x) + [R(x)/D(x)].My solution on the right reveals how I followed the aforementioned steps above to get the final answer. By the way, this problem was taken from my activity 1.
SPECIAL PRODUCTS
The third subtopic is special products. Special products is a mathematical term in which factors are combined to form products. It is called "special" because they do not need long solutions. The first is sum and difference of two binomials. This is one of the special pairs of binomial that produces a special product. Observe that the product of the sum and difference of these two binomials (x + y)(x − y) will result to the difference of the squares of the first and the last term, which is x2 − y2. The second is square of a binomial. The square of a binomial is the sum of: the square of the first term, twice the product of the two terms, and the square of the last term. The third is square of a trinomial. The square of a trinomial is the sum of: the square of the three terms, twice the product of the first and second term, twice the product of the first and last term, and twice the product of the second and last term. The fourth is cube of a binomial. The cube of a binomial is the sum of: the cube of the first term, thrice the product of the square of the first by the last term, thrice the product of the first term by the square of the last term, and the cube of the last term.
The third subtopic is special products. Special products is a mathematical term in which factors are combined to form products. It is called "special" because they do not need long solutions. The first is sum and difference of two binomials. This is one of the special pairs of binomial that produces a special product. Observe that the product of the sum and difference of these two binomials (x + y)(x − y) will result to the difference of the squares of the first and the last term, which is x2 − y2. The second is square of a binomial. The square of a binomial is the sum of: the square of the first term, twice the product of the two terms, and the square of the last term. The third is square of a trinomial. The square of a trinomial is the sum of: the square of the three terms, twice the product of the first and second term, twice the product of the first and last term, and twice the product of the second and last term. The fourth is cube of a binomial. The cube of a binomial is the sum of: the cube of the first term, thrice the product of the square of the first by the last term, thrice the product of the first term by the square of the last term, and the cube of the last term.
I picked an example for a square of a trinomial from my problem set #1. The formula on the first picture is just the summary of what has been described above. As stated, the square of a trinomial is the sum of: the square of the three terms, twice the product of the first and second term, twice the product of the first and last term, and twice the product of the second and last term. Using the formula, the problem was easily solved without utilizing the traditional method of multiplying the trinomial by itself.
FACTORING A POLYNOMIAL
The fourth subtopic is factoring a polynomial. A polynomial can be factored using different strategies or techniques depending upon the type of polynomial. It can be polynomials with common monomial factor in which it involves dividing the polynomial by the greatest common factor (GCF); difference of two squares or simply the difference of the squares of two terms is the product of the sum and difference of those terms [(x2 - y2) = (x + y)(x - y)]; sum and difference of two cubes such as x3 + y3 = (x + y)(x2 − xy + y2) and x3 − y3 = (x − y)(x2 + xy + y2); perfect square trinomials or the polynomials in which the first and last terms are perfect squares and the middle term is equal to twice the product of the square roots of the first and the last terms x2 + 2xy + y2 = (x + y)2 or x2 − 2xy + y2 = (x − y)2; and quadratic trinomials x2 + bx + c in which the factored form is (x + r)(x + s) where (r)(s) = c and r + s = b.
Different factoring techniques can be used for these types of polynomials. Aside from these techniques, I got interested with factoring trinomials ax2 + bx + c by grouping since the process was relatively new for me. Here are the steps to follow when factoring quadratic trinomials ax2 + bx + c where a≠1.1. Find the product of the coefficients a and c.2. Find the factors of ac whose sum is the coefficient of the middle term, b.3. Rewrite the polynomial by expressing the middle bx as a sum or difference of thefactors you found in Step 2.4. Factor by grouping.
The fourth subtopic is factoring a polynomial. A polynomial can be factored using different strategies or techniques depending upon the type of polynomial. It can be polynomials with common monomial factor in which it involves dividing the polynomial by the greatest common factor (GCF); difference of two squares or simply the difference of the squares of two terms is the product of the sum and difference of those terms [(x2 - y2) = (x + y)(x - y)]; sum and difference of two cubes such as x3 + y3 = (x + y)(x2 − xy + y2) and x3 − y3 = (x − y)(x2 + xy + y2); perfect square trinomials or the polynomials in which the first and last terms are perfect squares and the middle term is equal to twice the product of the square roots of the first and the last terms x2 + 2xy + y2 = (x + y)2 or x2 − 2xy + y2 = (x − y)2; and quadratic trinomials x2 + bx + c in which the factored form is (x + r)(x + s) where (r)(s) = c and r + s = b.
Different factoring techniques can be used for these types of polynomials. Aside from these techniques, I got interested with factoring trinomials ax2 + bx + c by grouping since the process was relatively new for me. Here are the steps to follow when factoring quadratic trinomials ax2 + bx + c where a≠1.1. Find the product of the coefficients a and c.2. Find the factors of ac whose sum is the coefficient of the middle term, b.3. Rewrite the polynomial by expressing the middle bx as a sum or difference of thefactors you found in Step 2.4. Factor by grouping.
In factoring this trinomial, one must follow the aforementioned four steps to get the correct factors since it belongs to ax2 + bx + c given that it has a coefficient of 6. The first thing I did is get the product of 6 and 45 which is 270. Then, I find the factors of 270 whose sum is equal to 59, subsequently they are 54 and 5. Then, I rewrite the polynomial through the factors 54 and 5. Lastly, I got the factors through grouping.
RADICAL EXPRESSIONS
Furthermore, this module also includes radical expressions. Radical expression is an expression containing a radical sign. Looking at the figure on the left , a is the coefficient, c is the radicand which is found inside the radical symbol and, b is the index that serves as the degree of the radicand c. The terms radical and radicand are both derived from the Latin word 'radix,' which means 'root'. The reason for this is that the root is the source of something (like the root of a word); if you square or cube a number, the number that it came from is the root, while the number itself (the radicand) grows from that root. The first usage of these terms was seen in England in the mid 1600s. They were first used in a book called An Introduction to Algebra by John Pell (Radical Expression: Definition & Examples, 2015).
Furthermore, this module also includes radical expressions. Radical expression is an expression containing a radical sign. Looking at the figure on the left , a is the coefficient, c is the radicand which is found inside the radical symbol and, b is the index that serves as the degree of the radicand c. The terms radical and radicand are both derived from the Latin word 'radix,' which means 'root'. The reason for this is that the root is the source of something (like the root of a word); if you square or cube a number, the number that it came from is the root, while the number itself (the radicand) grows from that root. The first usage of these terms was seen in England in the mid 1600s. They were first used in a book called An Introduction to Algebra by John Pell (Radical Expression: Definition & Examples, 2015).
Converting Expressions with Rational Exponents to Radical Form and Vice-Versa
Notice that when transforming expressions with rational exponents into radical form, the base serves as the radicand, and the denominator of the rational exponent is the index of the radical expression whereas the numerator is the exponent of the radicand.
Notice that when transforming expressions with rational exponents into radical form, the base serves as the radicand, and the denominator of the rational exponent is the index of the radical expression whereas the numerator is the exponent of the radicand.
Simplifying Radical Expressions
A radical expression is expressed in simplest form if it satisfies the following conditions:1. The radicand must not contain a perfect square (cube or etc., depending on the index) factor.2. The radicand must not contain a fraction.3. There must be no radical sign in the denominator of an expression.
There are three simple techniques to simplify radical expressions. The first one is simplifying radicals by reducing the radicand. It is simply removing the perfect nth powers of the radicand. The second one is simplifying radicals by rationalization. In case there is a radical sign in the denominator, rationalization is performed to eliminate the radical sign. The process is multiplying the original fraction by a particular expression so that there will be no longer a radical sign in the denominator. The last technique is simplifying radicals by reducing the index. In case that the radical expression contain high index, one can simplify it by transforming the radical expression into exponential form. Once it is in exponential form, reduce the rational exponent into lowest term then convert it back into radical form.
A radical expression is expressed in simplest form if it satisfies the following conditions:1. The radicand must not contain a perfect square (cube or etc., depending on the index) factor.2. The radicand must not contain a fraction.3. There must be no radical sign in the denominator of an expression.
There are three simple techniques to simplify radical expressions. The first one is simplifying radicals by reducing the radicand. It is simply removing the perfect nth powers of the radicand. The second one is simplifying radicals by rationalization. In case there is a radical sign in the denominator, rationalization is performed to eliminate the radical sign. The process is multiplying the original fraction by a particular expression so that there will be no longer a radical sign in the denominator. The last technique is simplifying radicals by reducing the index. In case that the radical expression contain high index, one can simplify it by transforming the radical expression into exponential form. Once it is in exponential form, reduce the rational exponent into lowest term then convert it back into radical form.
The expression given above has high index and was, by the way, taken from my activity 1. Subsequently, we can simplify it by reducing the index. As can be seen in my solution, the first I did is convert 256 into an exponential form which is 4 raised to 4. Then, I transformed the whole radical expression into an exponential form. I got the fractional or rational exponent 1/10. Then, I distributed this 1/10 to each of the exponent inside the parenthesis. Now, we can observe the lowest term of the exponents and so the expression's overall rational exponent. Lastly, I convert the exponential form back to radical form and is now simplified.
Operations of Radicals
To add or subtract radicals, the indices and radicands must be the same. If the radicands are not the same, you may simplify it first if possible in order to have like radicals. In dividing radical expression, apply the quotient rule for radicals which states that "the radical of a quotient is equal to the quotient of the radicals of the numerator and denominator". Further, in multiplying radical expression, there are three cases to take into consideration.
1. Indices are the same.When multiplying radicals with the same indices, apply product law of radical which states that "the radical of a product is equal to the radical of each of the factor".
2. Indices are different but radicands are the same.When multiplying radical expressions with different indices but with the same radicands, you need to transform the radical expressions into exponential form. Next, multiply the expressions by applying the product rule: xa ∙ xb =xa+b. Then, transform it back into radical form and simplify the result if possible.
3. Indices and radicands are different.When multiplying radical expressions with different indices and radicands, you need to transform the radical expressions into exponential form. Next, change the rational exponents of the exponential expressions into similar fractions. Then, combine the exponential expressions as a single product. Afterwards, convert it back to radical form, and simplify the result if possible.
To add or subtract radicals, the indices and radicands must be the same. If the radicands are not the same, you may simplify it first if possible in order to have like radicals. In dividing radical expression, apply the quotient rule for radicals which states that "the radical of a quotient is equal to the quotient of the radicals of the numerator and denominator". Further, in multiplying radical expression, there are three cases to take into consideration.
1. Indices are the same.When multiplying radicals with the same indices, apply product law of radical which states that "the radical of a product is equal to the radical of each of the factor".
2. Indices are different but radicands are the same.When multiplying radical expressions with different indices but with the same radicands, you need to transform the radical expressions into exponential form. Next, multiply the expressions by applying the product rule: xa ∙ xb =xa+b. Then, transform it back into radical form and simplify the result if possible.
3. Indices and radicands are different.When multiplying radical expressions with different indices and radicands, you need to transform the radical expressions into exponential form. Next, change the rational exponents of the exponential expressions into similar fractions. Then, combine the exponential expressions as a single product. Afterwards, convert it back to radical form, and simplify the result if possible.
To multiply the radical expressions above, first I had identified that the indices and radicands are different. By following the number 3 condition, I transformed the expressions into exponential forms. Then, as stated, I changed the fractional/rational exponents into one similar fraction 1/4. Subsequently, I combined the expressions into a single product in which all the expression will be under the fourth root radical sign. Then, in simplifying, I reduced the radicand by removing the perfect fourth which is 2 raised to 4. Lastly, I simplified the coefficient to get the final answer. Moreover, this problem was taken from my problem set 1.
Jess Marck Salazar - PS 1-Module 1.pdfPROBLEM SET #1 (37/40)
Salazar BSCE 1B (Activity 1).pdfACTIVITY #1 (100/100)
Salazar, Jess Marck - Quiz 1 Solutions -BSCE1B.pdfQUIZ #1 SOLUTIONS (35/35)
REFLECTION
I really enjoyed the whole module 1. I was surprised as I first read the material because the topics were already discussed in my high school years. Supposedly it will just be a review, yet I still learned new things. I learned a new technique of factoring a polynomial particularly factoring by grouping. Before, I was just using the trial-and-error method to get the factors of a quadratic polynomial. Now, the process became very easy. Further, I was really amazed with the material given to us because it was comprehensive that I was able to understand the flow of every lesson being discussed. The material was a very good reviewer. Actually, I already started making my own reviewer back then. Yet, I always feel uncomfortable for I know that there is a missing topic that I should have written. I always think there is a lacking topic. In short, I didn't finish it. But now, I was really glad that the subject Higher Mathematics for Engineers/Engineering Technologists gave me reviewers. Going back, all the computations in module 1, the problem set, activity and the quiz was quite easy and challenging at the same time. Nevertheless, I literally enjoyed finding the correct answer in every given problem. Meanwhile, in terms of teaching, I find it great that every detail is being discussed so as everyone could understand.
I really enjoyed the whole module 1. I was surprised as I first read the material because the topics were already discussed in my high school years. Supposedly it will just be a review, yet I still learned new things. I learned a new technique of factoring a polynomial particularly factoring by grouping. Before, I was just using the trial-and-error method to get the factors of a quadratic polynomial. Now, the process became very easy. Further, I was really amazed with the material given to us because it was comprehensive that I was able to understand the flow of every lesson being discussed. The material was a very good reviewer. Actually, I already started making my own reviewer back then. Yet, I always feel uncomfortable for I know that there is a missing topic that I should have written. I always think there is a lacking topic. In short, I didn't finish it. But now, I was really glad that the subject Higher Mathematics for Engineers/Engineering Technologists gave me reviewers. Going back, all the computations in module 1, the problem set, activity and the quiz was quite easy and challenging at the same time. Nevertheless, I literally enjoyed finding the correct answer in every given problem. Meanwhile, in terms of teaching, I find it great that every detail is being discussed so as everyone could understand.
REFERENCES:
https://wallpaperaccess.com/full/3280804.jpghttps://res.cloudinary.com/jerrick/image/upload/v1581025680/5e3c898fc935ad001e25fc42.jpghttps://cutewallpaper.org/21/color-background/Free-photo-color-background-Abstract-Bright-Closeup-.pnghttps://i.pinimg.com/564x/24/b7/31/24b7317b0cbfb62e564ee191fa24e887.jpghttps://24wallpapers.com/app-gateway/wallpaper-uploads/wallpapers/legacyUploads/wi89124833a91-caf7-4c3e-aad5-42da99b7dba92079665946.jpghttps://kinetikgroup.co/wp-content/uploads/2020/05/Beulah-International-MelbourneCBD.jpghttps://hips.hearstapps.com/hmg-prod.s3.amazonaws.com/images/downtown-shanghai-royalty-free-image-685014464-1567127429.jpghttps://math24seven.weebly.com/uploads/4/8/2/8/4828203/editor/properties-of-exponents.jpg?1507653167https://www.algebra-class.com/binomial.htmlhttps://study.com/academy/lesson/radical-expression-definition-examples-quiz.htmlhttps://study.com/cimages/multimages/16/study_radical_028453003183610235973.pnghttps://qph.fs.quoracdn.net/main-qimg-58370f7001d34ab10fb1cdd42cd7b5b9
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