9.3: Multiplication of Monomials by Polynomials
Did you know that the formula for the volume of a pyramid is
Guidance
When multiplying polynomials together, we must remember the exponent rules we learned in previous Concepts, such as the Product Rule. This rule says that if we multiply expressions that have the same base, we just add the exponents and keep the base unchanged. If the expressions we are multiplying have coefficients and more than one variable, we multiply the coefficients just as we would any numbers. We also apply the product rule on each variable separately.
Example A
Multiply
Solution:
Multiplying a Polynomial by a Monomial
This is the simplest of polynomial multiplications. Problems are like the one above.
Example B
Multiply the following monomials.
(a)
(c)
(d)
Solutions:
(a)
(c)
(d)
To multiply monomials, we use the Distributive Property.
Distributive Property: For any expressions
This property can be used for numbers as well as variables. This property is best illustrated by an area problem. We can find the area of the big rectangle in two ways.
One way is to use the formula for the area of a rectangle.
The area of the big rectangle can also be found by adding the areas of the two smaller rectangles.
This means that
In general, if we have a number or variable in front of a parenthesis, this means that each term in the parenthesis is multiplied by the expression in front of the parenthesis.
Example C
Multiply
Solution:
Guided Practice
Multiply
Solution:
Multiply the monomial by each term inside the parenthesis:
Practice
Multiply the following monomials.

(2x)(−7x) 
4(−6a) 
(−5a2b)(−12a3b3) 
(−5x)(5y) 
y(xy4) 
(3xy2z2)(15x2yz3)
Multiply and simplify.

x8(xy3+3x) 
2x(4x−5) 
6ab(−10a2b3+c5) 
9x3(3x2−2x+7) 
−3a2b(9a2−4b2)
Mixed Review
 Give an example of a fourth degree trinomial in the variable
n .  Find the next four terms of the sequence
1,32,94,288,…  Reece reads three books per week.
 Make a table of values for weeks zero through six.
 Fit a model to this data.
 When will Reece have read 63 books?
 Write 0.062% as a decimal.
 Evaluate
ab(a+b4) whena=4 andb=−3 .  Solve for
s :3s(3+6s)+6(5+3s)=21s .
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Here you'll learn how to multiply monomials by polynomials.