Polynomials
Multiplication of Polynomials
Multiplication of a Polynomial by a Monomial
To multiply a polynomial by a monomial, use the distributive property: multiply each term of the polynomial by the monomial. This involves multiplying coefficients and adding exponents of the appropriate variables.
Example 1:
3y
2(12y
3 -6y
2 + 5y - 1) =
?
= 3y
2(12y
3) + (3y
2)(- 6y
2) + (3y
2)(5y) + (3y
2)(- 1)
= (3)(12)y
2+3 + (3)(- 6)y
2+2 + (3)(5)y
2+1 + (3)(- 1)y
2
= 36y
5 -18y
4 +15y
3 -3y
2
Example 2:
-4x
3
y(- 2y
2 + xy - x + 9) =
?
= - 4x
3
y(- 2y
2) + (- 4x
3
y)(xy) + (- 4x
3
y)(- x) + (- 4x
3
y)(9)
= (- 4)(- 2)x
3
y
1+2 + (- 4)x
3+1
y
1+1 + (- 4)(- 1)x
3+1
y + (- 4)(9)x
3
y
= 8x
3
y
3 -4x
4
y
2 +4x
4
y - 36x
3
y
Multiplication of Binomials
To multiply a binomial by a binomial--
(a + b)(c + d )
, where
a
,
b
,
c
, and
d
are terms--use the distributive property twice.
First, treat the second binomial as a single term and distribute over
the first binomial:
| (a + b)(c + d )= a(c + d )+ b(c + d ) |
Next, use the distributive property over the second binomial:
| a(c + d )+ b(c + d )= ac + ad + bc + bd |
At this point, there should be 4 terms in the answer -- every combination of a term of the first binomial and a term of the second binomial. Simplify the answer by combining like terms.
We can use the word FOIL to remember how to multiply two binomials (a + b)(c + d ) :
- Multiply their First terms. (ac)
- Multiply their Outside terms. (ad )
- Multiply their Iinside terms. (bc)
- Multiply their Last terms. (bd )
- Finally, add the results together: ac + ad + bc + bd . Combine like terms.
Example 1.
(xy + 6)(x + 2y) =
?
= (xy)(x) + (xy)(2y) + (6)(x) + (6)(2y)
= x
2
y + 2xy
2 + 6x + 12y
Example 2.
(3x
2 +7)(4 - x
2) =
?
= (3x
2)(4) + (3x
2)(- x
2) + (7)(4) + (7)(- x
2)
= 12x
2 -3x
4 +28 - 7x
2
= - 3x
4 + (12 - 7)x
2 + 28
= - 3x
4 +5x
2 + 28
Example 3:
(y - x)(- 4y - 3x) =
?
= (y)(- 4y) + (y)(- 3x) + (- x)(- 4y) + (- x)(- 3x)
= - 4y
2 -3xy + 4xy + 3x
2
= 3x
2 + (- 3 + 4)xy - 4y
2
= 3x
2 + xy - 4y
2
Multiplication of Polynomials
The strategy for multiplying two polynomials in general is similar to
multiplying two binomials. First, treat the second polynomial as a
single term, and distribute
over the first term:
| (a + b + c)(d + e + f )= a(d + e + f )+ b(d + e + f )+ c(d + e + f ) |
Next, distribute over the second polynomial:
| a(d + e + f )+ b(d + e + f )+ c(d + e + f )= ad + ae + af + bd + be + bf + cd + ce + cf |
At this point, the number of terms in the answer should be the number in the first polynomial times the number in the second polynomial--every combination of a term of the first polynomial and a term of the second polynomial. Since there are 3 terms in each polynomial in this example there should be 3(3) = 9 terms in our answer so far. If the first polynomial had 4 terms and the second had 5 , there would be 4(5) = 20 terms in the answer so far.
Finally, since the the terms in such a product of polynomials are often highly redundant (many have the same variables and exponents), it is important to combine like terms.
Example 1:
(x
2 -2)(3x
2 - 3x + 7) =
?
= x
2(3x
2 -3x + 7) - 2(3x
2 - 3x + 7)
= x
2(3x
2) + x
2(- 3x) + x
2(7) - 2(3x
2) - 2(- 3x) - 2(7)
(
6
terms)
= 3x
4 -3x
3 +7x
2 -6x
2 + 6x - 14
= 3x
4 -3x
3 + (7 - 6)x
2 + 6x - 14
= 3x
4 -3x
3 + x
2 + 6x - 14
Example 2:
(x
2 + x + 3)(2x
2 - 3x + 1) =
?
= x
2(2x
2 -3x + 1) + x(2x
2 -3x + 1) + 3(2x
2 - 3x + 1)
= x
2(2x
2) + x
2(- 3x) + x
2(1) + x(2x
2) + x(- 3x) + x(1) + 3(2x
2) + 3(- 3x) + 3(1)
(
9
terms)
= 2x
4 -3x
3 + x
2 +2x
3 -3x
2 + x + 6x
2 - 9x + 3
= 2x
4 + (- 3 + 2)x
3 + (1 - 3 + 6)x
2 + (1 - 9)x + 3
= 2x
4 - x
3 +4x
2 - 8x + 3
Note: To check your answer, pick a value for the variable and evaluate both the original expression and your answer--they should be the same.
