4.2 Multiplying a Polynomial and a Monomial

To multiply a monomial by a polynomial, we can use the Distributive Property.

Table 4.2.1
Distributive Property
If a, b, and c are real numbers, then $a(b+c)=ab+ac$

Example 1

Multiply: $4\left(x+3\right)$

Table 4.2.2
Steps	Solution
Distribute.	$4 \bullet x + 4 \bullet 3$
Simplify.	$4x + 12$

Practice 1

Multiply.

a) $5\left(x+7\right)$

b) $-2y\left(4{y}^{2}+3y-5\right)$

c) $7x\left(2x+y\right)$

d) $2{x}^{3}\left({x}^{2}-8x+1\right)$

e) $\left(x+3\right)p$

f) $\left(-y-2\right)5$

Practice 2

Multiply and simplify.

a) $3 + 2(4x - 5)$

b) $5a - 3(2a + 4b)$

c) $2(3x - 5) + 5(x + 6)$

d) $4(2a + 3b - 1) - 3(a - 2b + 3)$

e) $4b^2 + 5 - 3b(2b^2 - 3b + 1)$

Practice 3

Write a simplified expression for the perimeter of a triangle with side lengths 4x + 2, 3x – 5, and 10x + 1. Draw a diagram.

Practice 4

Write a simplified expression for the area of a square with side lengths 5a³. Draw a diagram.

Practice 5

Write a simplified expression for the area of a rectangle with a length of 4x + 2 and a width of 3x. Draw a diagram.

Homework

Multiply.

a) $3\left(y+13\right)$

b) $x\left(x-7\right)$

c) $d\left(d-11\right)$

d) $2p\left(6p+r\right)$

e) $-3y\left(5{y}^{2}+8y-7\right)$

f) $4{x}^{2}\left(2{x}^{2}-3x+5\right)$

g) $4x\left(3{x}^{2}-5x+3\right)$

h) $-6{a}^{3}\left(3{a}^{2}-2a+6\right)$

i) $\left(x+8\right)p$

j) $\left(a+4\right)p$
Multiply and simplify.

a) $4 + 3(5x - 7)$

b) $2m - 4(3m + 2n)$

c) $5(2x + 3) + 3(x - 4)$

d) $6(2a - 3b) - 2(4a + b)$

e) $7(3y - 2) + 5(2y + 4)$

f) $3(4x - 5y + 6) + 2(x + y - 7)$

g) $3(2p + 4q - 5) - 2(3p - 2q + 8)$

h) $5(3x + 2y) - 4(x - 6y)$
Multiply and simplify.

a) $2(4x^2 - 5x) + 3(2x^2 + 3x)$

b) $4a(2a + 3b^2) - 2(3a - 2b^2)$

c) $5x + 3(3x^2 - 2x + 4) + 2(x - 3)$

d) $6y(2y^2 + 3y) - 4(5y^2 - 2)$

e) $3(2x^2 - 3y + 4) + 2x^2 + 5y - 6$

f) $2a(4a^2 + 3b - 1) - 3a + 6b^2 - 9$

g) $3(2p^2 + 4q - 5) - 2(3p - 2q^2 + 8)$

h) $5x + 4(x^2 + 4) - 2(x^2 - 3x)$
Write a simplified expression to represent the perimeter of the following shapes. Draw a diagram.

a) A triangle with side lengths –2n + 7, 4n – 1, and 8n + 12

b) A square with side lengths 6m²

c) A rectangle with a length of 4q – 1 and a width of 2q
Write a simplified expression to represent the area of the following shapes. Draw a diagram.

a) A square with side lengths 5a³

b) A rectangle with a length of 2b – 6 and a width of b

c) A rectangle with a length of –3h + 1 and a width of –5h

Answers

1.

a) $3y+39$	b) ${x}^{2}-7x$
c) ${d}^{2}-11d$	d) $12{p}^{2}+2pr$
e) $-15{y}^{3}-24{y}^{2}+21y$	f) $8{x}^{4}-12{x}^{3}+20{x}^{2}$
g) $12{x}^{3}-20{x}^{2}+12x$	h) $-18{a}^{5}+ 12{a}^{4}-36{a}^{3}$
i) $xp+8p$	j) $ap+4p$

2.

a) $15x - 17$	b) $-10m - 8n$
c) $13x + 3$	d) $4a - 20b$
e) $31y + 6$	f) $14x - 13y + 4$
g) $16q - 31$	h) $11x + 34y$

3.

a) $14x^2 - x$	b) $8a^2 + 12ab^2 - 6a + 4b^2$
c) $9x^2 + x + 6$	d) $12y^3 - 2y^2 + 8$
e) $8x^2 - 4y + 6$	f) $8a^3 + 6ab - 5a + 6b^2 - 9$
g) $6p^2 - 6p + 4q^2 + 12q - 31$	h) $2x^2 + 11x + 16$

4.

a) $10n + 18$

b) $24m^2$

c) $12q - 2$

5.

a) $25a^6$

b) $2b^2 - 6b$

c) $15h^2 - 5h$

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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Math 10C Workbook Copyright © 2026 by Leanne Thompson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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