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