To add and subtract polynomials, you must recall the difference between like and unlike terms.
Study the examples of like terms and unlike terms.
Like terms Unlike terms
2, -5, 7.5 3, y
x, 8x, 3x2,
3xy2, -4xy2 3xy2, 3x2y
Determine which of the following are pairs of like terms by placing a check before the number.
_____1. 25b, 25b2 _____4. a4b2c, -11a4b2c
_____2. –3x3y2, 2x3y2 _____5. –0.4,
_____3. –0.5c, _____6. 3x5y, 3xy5
If you check 2, 3, 4 and 5, then you’re correct!
Like terms or similar terms are monomials that contain the same literal coefficients, that is, the terms have exactly the same variables and exponents.
A polynomial is in the simplest form when all like terms are combined.
Study and learn from the illustration below.
Simplify
1. 11t + 7t = (11+7)t
=18t
2. 2xy-5xy+4xy= (2-5+4)xy
= xy
3. 2x2-5x+6x-x2+11= (2x2-x2) + (-5x+6x) +11 Group like terms
= (2-1)x2 +(-5+6)x + 11
= x2 + x + 11
Distributive Property: a(b+c) = ab + bc
In simplifying polynomials, combine like terms by adding or subtracting their numerical coefficients and multiplying the result by the common literal coefficient.
ADDITION OF POLYNOMIALS
Illustrative Examples
1. Add x2-2x + 3 and 4x2+x-2
(x2-2x + 3) + (4x2+x-2) = (x2+4x2) + (-2x + x) + (3-2) Grouping like terms
= (1+4)x2 + (-2+1) x +(3-2) Distributive property
= 5x2 – x + 1
2. Add: ( 3 – 2x + x2) + (6x2 + 5x –4)
A vertical arrangement may be used to add polynomials. Like terms are written in the same column.
x2 – 2x + 3
+ 6x2 + 5x - 4
7x2 + 3x –1
SUBTRACTION OF POLYNOMIALS
Since subtraction is defined as addition of the opposite or additive inverse, subtraction is very similar to addition of polynomials.
Illustrative Examples
1. Subtract: (4x2-4x) – (7x2 – 3x)
(4x2-4x) – (7x2 – 3x) = ( 4x2 - 4x) + ( -7x2 + 3x) Add the additive inverse
= (4x2 – 7x2) + (-4x + 3x) Grouping like terms
= (4-7)x2 + (-4+3)x Distributive property
= -3x2 – x
2. Subtract 4a3 –2a2 + 6a – 5 from 6a3 –5a2 + 11a + 8
The minuend is 6a3 –5a2 + 11a + 8
The subtrahend is 4a3 –2a2 + 6a – 5
The mere identification of the minuend and the subtrahend facilitate accuracy in subtracting polynomials.
Subtracting polynomials vertically,
6a3 –5a2 + 11a + 8 → 6a3 –5a2 + 11a + 8
- ( 4a3 –2a2 + 6a – 5) → + -4a3 +2a2 –6a + 5
2a3 – 3a2 + 5a + 13
A polynomial is in the simplest form when all like terms are combined. Like terms or similar terms are monomials that contain the same literal coefficients, that is, the terms have exactly the same variables and exponents.
Simplify
1. 11t + 7t = (11+7)t
=18t
2. 2xy-5xy+4xy= (2-5+4)xy
= xy
3. 2x2-5x+6x-x2+11= (2x2-x2) + (-5x+6x) +11 Group like terms
= (2-1)x2 +(-5+6)x + 11
= x2 + x + 11
Distributive Property: a(b+c) = ab + bc
In simplifying polynomials, combine like terms by adding or subtracting their numerical coefficients and multiplying the result by the common literal coefficient.
4. Add x2-2x + 3 and 4x2+x-2
(x2-2x + 3) + (4x2+x-2) = (x2+4x2) + (-2x + x) + (3-2) Grouping like terms
= (1+4)x2 + (-2+1) x +(3-2) Distributive property
= 5x2 – x + 1
5. Subtract:: (4x2-4x) – (7x2 – 3x)
(4x2-4x) – (7x2 – 3x) = ( 4x2 - 4x) + ( -7x2 + 3x) Add the additive inverse
= (4x2 – 7x2) + (-4x + 3x) Grouping like terms
= (4-7)x2 + (-4+3)x Distributive property
= -3x2 – x