What if you had a monomial and polynomial like
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CK12 Foundation: Multiplying a Polynomial by a Monomial
Guidance
Just as we can add and subtract polynomials, we can also multiply them. The Distributive Property and the techniques you’ve learned for dealing with exponents will be useful here.
Multiplying a Polynomial by a Monomial
When multiplying polynomials, we must remember the exponent rules that we learned in the last chapter. Especially important is the product rule:
If the expressions we are multiplying have coefficients and more than one variable, we multiply the coefficients just as we would any number and we apply the product rule on each variable separately.
Example A
Multiply the following monomials.
a)
b)
c)
d)
Solution
a)
b)
c)
d)
To multiply a polynomial by a monomial, we have to use the Distributive Property. Remember, that property says that
Example B
Multiply:
a)
b)
c)
Solution
a)
b)
c)
Notice that when we use the Distributive Property, the problem becomes a matter of just multiplying monomials by monomials and adding all the separate parts together.
Example C
Multiply:
a)
b)
Solution
a)
b)
Watch this video for help with the Examples above.
CK12 Foundation: Multiplying a Polynomial by a Monomial
Vocabulary

Distributive Property: For any expressions
a, b , andc ,a(b+c)=ab+ac .
Guided Practice
Multiply
Solution:
Multiply the monomial by each term inside the parenthesis:
Explore More
Multiply the following monomials.

(2x)(−7x) 
(10x)(3xy) 
(4mn)(0.5nm2) 
(−5a2b)(−12a3b3) 
(3xy2z2)(15x2yz3)
Multiply and simplify.

17(8x−10) 
2x(4x−5) 
9x3(3x2−2x+7) 
3x(2y2+y−5) 
10q(3q2r+5r) 
−3a2b(9a2−4b2)