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
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 ) |
a(c + d )+ b(c + d )= ac + ad + bc + bd |
We can use the word FOIL to remember how to multiply two binomials (a + b)(c + d ) :
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}
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 ) |
a(d + e + f )+ b(d + e + f )+ c(d + e + f )= ad + ae + af + bd + be + bf + cd + ce + cf |
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.