Additional Notes: Recall from unit 9 that when bases are being divided, you keep the base and subtract the exponents. Now we are going to include expressions that contain a coefficient (remember, that is a number attached to, and in front of, a variable.) We are going to divide the COEFFICIENTS, keep the base and add the exponents. Remember to use caution with negative exponents.

• 49x^8 / 7x^5 = 7x^3 The exponents are attached to the base of x, so x is the base we are going to keep. Just like before, since the x's are being divided, we have to subtract the exponents. Those 2 facts are the same as last unit. Now we also have a 49 and a 7 in front of each x. The 49 and 7 are not the base since they have no exponent, therefore it is just a number. When we have numbers with a / sign, it means to divide them.

Divide the coefficients: 49/7 = 7 Keep the base of x and subtract the exponents: 8-5 = 3. Combine the answers to yield 7x^3

• 12x^8 / 3x^5 = 4x^3 because Divide the coefficients: 12/3 = 4 Keep the base of x and subtract the exponents: 8-5 = 3.
• 20x^-8 / 4x^5 = 5/x^13 because Divide the coefficients: 20/4 = 5 Keep the base of x and subtract the exponents: -8-5=-13 giving an answer of 5x^13 we would then get rid of the negative exponent by doing the reciprocal: 5/x^13 (We only do the reciprocal to BASES with negative exponents. 5 does not get the -13 exponent)
• 120x^8 / 12x^-5 = 10^13 because Divide the coefficients: 120/12 = 10 Keep the base of x and subtract the exponents: 8- -5= 8 + 5 = 13
• Remember, when coefficients don't divide, simplify the fraction (what can you divide each number by?)

12x^10 / 10x^7 = 6x^3/5 (six x to the third divided by five) Let's work backwards: Keep the base of x and subtract the exponents: 10-7=3. When we go to divide the coefficients 12/10, it does not divide nicely. We NEVER write as mixed numbers when variables and exponents are involved. That means that we have to simplify the improper fraction into another improper fraction. Both 12 and 10 are divisible by 2. That gives 6/5. Since x has a positive exponent, it stays in the numerator with the 6. All answers are written as a SINGLE FRACTION.

Additional Notes: Recall that when a number is attached to parentheses by multiplication, we apply the Distributive Property. When it is attached by division, we distribute the division to each term.

• (30x^8 + 15x^3 - 5x) / 5x write out as 3 separate quotients:

30x^8/5x +15x^3/5x -5x/5x then follow the division rules for each of the 3 sections (don't forget + and - signs)

6x^7 +3x^2 -1 then put together with appropriate signs (recall anything divided by itself is 1. Many people forget the 1!)

6x^7 + 3x^2 - 1

Additional Notes:

• Anything (except 0) to the zero power is 1
• To make a negative exponent positive, do the reciprocal. Then follow multiplication or division rules.

F 5/8 Mixed Practice

Today you will continue practicing rules.

Log into Delta Math and complete assignment "8RDL Polynomials 4"