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# Multiplying Polynomials Practice Questions

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Updated on Sep 23, 2011

## Introduction

This set of practice questions will present problems for you to solve in the multiplication of polynomials. Specifically, you will practice solving problems multiplying a monomial (one term) and a polynomial, multiplying binomials (expressions with two terms), and multiplying a trinomial and a binomial.

## Tips for Multiplying Polynomials

When multiplying a polynomial by a monomial, you use the distributive property of multiplication to multiply each term in the polynomial by the monomial.

a(b + c + d + e) = ab + ac + ad + ae

When multiplying a binomial by a binomial, you use the mnemonic FOIL to remind you of the order with which you multiply terms in the binomials.   (a + b)(c + d)

F is for first. Multiply the first terms of each binomial.             ([a] + b)([c] + d) gives the term ac.

O is for outer. Multiply the outer terms of each binomial.         ([a] + b)(c + [d]) gives the term ad.

I is for inner. Multiply the inner terms of each binomial.           (a + [b])([c] + d) gives the term bc.

L is for last. Multiply the last terms of each binomial.               (a + [b])(c + [d]) gives the term bd.

Then you combine the terms.                                                   ac + ad + bc + bd

Multiplying a trinomial by a binomial is relatively easy. You proceed similarly to the way you would when using the distributive property of multiplication. Multiply each term in the trinomial by the first and then the second term in the binomial. Then add the results.

(a + b)(c + d + e) = (ac + ad + ae) + (bc + bd + be)

## Practice Questions

Multiply the following polynomials.

1. x(5x + 3y – 7)
2. 2a(5a2 – 7a + 9)
3. 4bc(3b2c + 7b – 9c + 2bc2 – 8)
4. 3mn(4m + 6n + 7mn2 – 3m2n)
5. (x + 3)(x + 6)
6. (x – 4)(x – 9)
7. (2x + 1)(3x – 7)
8. (x + 2)(x – 3y)
9. (7x + 2y)(2x – 4y)
10. (5x + 7)(5x – 7)
11. (28x + 7)( – 11)
12. (3x2 + y2)(x2 – 2y2)
13. (4 + 2x2)(9 – 3x)
14. (2x2 + y2)(x2y2)
15. (x + 2)(3x2 – 5x + 2)
16. (2x – 3)(x2 + 3x2 – 4x)
17. (4a + b)(5a2 + 2abb2)
18. (3y – 7)(6y2 – 3y + 7)
19. (3x + 2)(3x2 – 2x – 5)
20. (x + 2)(2x + 1)(x – 1)
21. (3a – 4)(5a + 2)(a + 3)
22. (2n – 3)(2n + 3)(n + 4)
23. (5r – 7)(3r4 + 2r2 + 6)
24. (3x2 + 4)(x – 3)(3x2 – 4)

Numerical expressions in parentheses like this [ ] are operations performed on only part of the original expression. The operations performed within these symbols are intended to show how to evaluate the various terms that make up the entire expression.

Expressions with parentheses that look like this ( ) contain either numerical substitutions or expressions that are part of a numerical expression. Once a single number appears within these parentheses, the parentheses are no longer needed and need not be used the next time the entire expression is written.

When two pair of parentheses appear side by side like this ( )( ), it means that the expressions within are to be multiplied.

Sometimes parentheses appear within other parentheses in numerical or algebraic expressions. Regardless of what symbol is used, ( ), { }, or [ ], perform operations in the innermost parentheses first and work outward.

Underlined expressions show the simplified result.

 1. Multiply each term in the trinomial by x. x(5x) + x(3y) – x(7) Simplify terms.
 2. Multiply each term in the trinomial by 2a. 2a(5a2) – 2a(7a) + 2a(9) Simplify terms.
 3. Multiply each term in the polynomial by 4bc. 4bc(3b2c) + 4bc(7b) – 4bc(9c) + 4bc(2bc2) – 4bc(8) Simplify terms.
 4. Multiply each term in the polynomial by 3mn. 3mn(–4m) + 3mn(6n) + 3mn(7mn2) – 3mn(3m2n) Simplify terms.
 5. Multiply each term in the polynomial by 4x. Simplify terms. Use the distributive property in the numerator of the fourth term. When similar factors, or bases, are being divided, subtract the exponent in the denominator from the exponent in the numerator. 36x4 + 12x1–2 – 4x5 + 24x2–2 – 4x1–2 Simplify operations in the exponents. 36x4 + 12x–1 – 4x5 + 24x0 – 4x–1 Use the associative property of addition. 36x4 + 12x–1 – 4x–1 – 4x5 + 24x0 Combine like terms. 36x4 + 8x–1 – 4x5 + 24x0 A base with a negative exponent in the numerator is equivalent to the same variable or base in the denominator with the inverse sign for the exponent. A variable to the power of zero equals 1. Simplify and put in order.
 6. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([x] + 3)([x] + 6) x2 Multiply the outer terms in each binomial. ([x] + 3)(x + [6]) +6x Multiply the inner terms in each binomial. (x + [3])([x] + 6) +3x Multiply the last terms in each binomial. (x + [3])(x + [6]) +18 Add the products of FOIL together. x2 + 6x + 3x + 18 Combine like terms.
 7. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([x] – 4)([x] – 9) x2 Multiply the outer terms in each binomial. ([x] – 4)(x – [9]) –9x Multiply the inner terms in each binomial. (x – [4])([x] – 9) –4x Multiply the last terms in each binomial. (x – [4])(x – [9]) +36 Add the products of FOIL together. x2 – 9x – 4x + 36 Combine like terms.
 8. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([2x] + 1)([3x] – 7) 6x2 Multiply the outer terms in each binomial. ([2x] + 1)(3x – [7]) –14x Multiply the inner terms in each binomial. (2x + [1])([3x] – 7) +3x Multiply the last terms in each binomial. (2x + [1])(3x – [7]) –7 Add the products of FOIL together. 6x2 – 14x + 3x – 7 Combine like terms.
 9. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([x] + 2)([x] – 3y) x2 Multiply the outer terms in each binomial. ([x] + 2)(x – [3y]) –3xy Multiply the inner terms in each binomial. (x + [2])([x] – 3y) +2x Multiply the last terms in each binomial. (x + [2])(x – [3y]) –6y Add the products of FOIL together.
 10. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([7x] + 2y)([2x] – 4y) 14x2 Multiply the outer terms in each binomial. ([7x] + 2y)(2x – [4y]) –28xy Multiply the inner terms in each binomial. (7x + [2y])([2x] – 4y) +4xy Multiply the last terms in each binomial. (7x + [2y])(2x – [4y]) –8y2 Add the products of FOIL together. 14x2 – 28xy + 4xy – 8y2 Combine like terms.
 11. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([5x] + 7)([5x] – 7) 25x2 Multiply the outer terms in each binomial. ([5x] + 7)(5x – [7]) –35x Multiply the inner terms in each binomial. (5x + [7])([5x] – 7) +35x Multiply the last terms in each binomial. (5x + [7] (5x – [7]) –49 Add the products of FOIL together. 25x2 – 35x + 35x – 49 Combine like terms.
 12. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([28x] + 7)([] – 11) 4x2 Multiply the outer terms in each binomial. ([28x] + 7)( – [11]) –308x Multiply the inner terms in each binomial. (28x + [7])([] – 11) +x Multiply the last terms in each binomial. (28x + [7])( – [11]) –77 Add the products of FOIL together. 4x2 – 308x + x– 77 Combine like terms.
 13. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([3x2] + y2)([x2] – 2y2) 3x4 Multiply the outer terms in each binomial. ([3x2] + y2)(x2 – [2y2]) –6x2y2 Multiply the inner terms in each binomial. (3x2 + [y2])([x2] – 2y2 +x2y2 Multiply the last terms in each binomial. (3x2 + [y2])(x2 – [2y2]) –2y4 Add the products of FOIL together. 3x4 – 6x2y2 + x2y2 – 2y4 Combine like terms.
 14. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([4] + 2x2)([9] – 3x) +36 Multiply the outer terms in each binomial. ([4] + 2x2)(9 – [3x]) –12x Multiply the inner terms in each binomial. (4 + [2x2])([9] – 3x) +18x2 Multiply the last terms in each binomial. (4 + [2x2])(9 – [3x]) –3x3 Add the products of FOIL together. 36 – 12x + 18x2 – 3x3 Combine like terms.
 15. Use FOIL to multiply binomials. Multiply the first terms in each binomial. ([2x2] + y2)([x2] – y2) 2x4 Multiply the outer terms in each binomial. ([2x2] + y2)(x2 – [y2]) –2x2y2 Multiply the inner terms in each binomial. (2x2 + [y2])([x2] – y2) +x2y2 Multiply the last terms in each binomial. (2x2 + [y2])(x2 – [y2]) –y4 Add the products of FOIL together. 2x4 – 2x2y2 + x2y2 – y4 Combine like terms.
 16. Multiply the trinomial by the first term in the binomial, x. [x(3x2 – 5x + 2)] [x(3x2) – x(5x) + x(2)] Simplify terms. [3x3 – 5x2 + 2x] Multiply the trinomial by the second term in the binomial, 2. [2(3x2 – 5x + 2)] [2(3x2) – 2(5x) + 2(2)] Simplify terms. [6x2 – 10x + 4] Add the results of multiplying by the terms in the binomial together. [3x3 – 5x2 + 2x] + [6x2 – 10x + 4] Use the commutative property of addition. 3x3 – 5x2 + 6x2 + 2x – 10x + 4 Combine like terms.
 17. Multiply the trinomial by the first term in the binomial, 2x. [2x(x3 + 3x2 – 4x)] Use the distributive property of multiplication. [2x(x3) + 2x(3x2) – 2x(4x)] Simplify terms. [2x4 + 6x3 – 8x2] Multiply the trinomial by the second term in the binomial, –3. [–3(x3 + 3x2 – 4x)] Use the distributive property of multiplication. [–3(x3) – 3(3x2) – 3(–4x)] Simplify terms. [–3x3 – 9x2 + 12x] Add the results of multiplying by the terms in the binomial together. [2x4 + 6x3 – 8x2] + [–3x3 – 9x2 + 12x)] Use the commutative property of addition. 2x4 + 6x3 – 3x3 + 8x2 – 9x2 + 12x Combine like terms.
 18. Multiply the trinomial by the first term in the binomial, 4a. [4a(5a2 + 2ab – b2)] Use the distributive property of multiplication. [4a(5a2) + 4a(2ab) – 4a(b2)] Simplify terms. [20a3 + 8a2b – 4ab2] Multiply the trinomial by the second term in the binomial, b. [b(5a2 + 2ab – b2)] Use the distributive property of multiplication. [b(5a2 + b(2ab) – b(b2)] Simplify terms. [5a2b + 2ab2 – b3] Add the results of multiplying by the terms in the binomial together. [20a3 + 8a2b – 4ab2] + [5a2b + 2ab2 – b3)] Use the commutative property of addition. 20a3 + 8a2b + 5a2b – 4ab2 + 2ab2 – b3 Combine like terms.
 19. Multiply the trinomial by the first term in the binomial, 3y. [3y(6y2 – 3y + 7)] Use the distributive property of multiplication. [3y(6y2) – 3y(3y) + 3y(7)] Simplify terms. [18y3 – 9y2 + 21y] Multiply the trinomial by the second term in the binomial, –7. [–7(6y2 – 3y + 7)] Use the distributive property of multiplication. [–7(6y2) – 7(–3y) – 7(7)] Simplify terms. [–42y2 + 21y – 49] Add the results of multiplying by the terms in the binomial together. [18y3 – 9y2 + 21y] + [–42y2 + 21y – 49] Use the commutative property of addition. 18y3 – 9y2 – 42y2 + 21y + 21y – 49 Combine like terms.
 20. Multiply the trinomial by the first term in the binomial, 3x. [3x(3x2 – 2x – 5)] Use the distributive property of multiplication. [3x(3x2) – 3x(2x) – 3x(5)] Simplify terms. [9x3 – 6x2 – 15x] Multiply the trinomial by the second term in the binomial, 2. [2(3x2 – 2x – 5)] Use the distributive property of multiplication. [2(3x2) – 2(2x) – 2(5)] Simplify terms. [6x2 – 4x – 10] Add the results of multiplying by the terms in the binomial together. [9x3 – 6x2 – 15x] + [6x2 – 4x – 10] Use the commutative property of addition. 9x3 – 6x2 + 6x2 – 15x – 4x – 10 Combine like terms.
 21. Multiply the first two parenthetical terms in the expression using FOIL. Multiply the first terms in each binomial. ([x] + 2)([2x] + 1) 2x2 Multiply the outer terms in each binomial. ([x] + 2)(2x + [1]) +x Multiply the inner terms in each binomial. (x + [2])([2x] + 1) +4x Multiply the last terms in each binomial. (x + [2])(2x + [1]) +2 Add the products of FOIL together. 2x2 + x + 4x + 2 Combine like terms. 2x2 + 5x + 2 Multiply the resulting trinomial by the last binomial in the original expression. (x – 1)(2x2 + 5x + 2) Multiply the trinomial by the first term in the binomial, x. [x(2x2 + 5x + 2)] Use the distributive property of multiplication. [x(2x2) + x(5x) + x(2)] Simplify terms. [2x3 + 5x2 + 2x] Multiply the trinomial by the second term in the binomial, –1. [–1(2x2 + 5x + 2)] Use the distributive property of multiplication. [–2x2 – 5x – 2] Add the results of multiplying by the terms in the binomial together. [2x3 + 5x2 + 2x] + [–2x2 – 5x – 2] Use the commutative property of addition. 2x3 + 5x2 – 2x2 + 2x – 5x – 2 Combine like terms.
 22. Multiply the first two parenthetical terms in the expression using FOIL. Multiply the first terms in each binomial. ([3a] – 4)([5a] + 2) 15a2 Multiply the outer terms in each binomial. ([3a] – 4)(5a + [2]) +6a Multiply the inner terms in each binomial. (3a – [4])([5a] + 2) –20a Multiply the last terms in each binomial. (3a – [4])(5a + [2]) –8 Add the products of FOIL together. 15a2 + 6a – 20a – 8 Combine like terms. 15a2 – 14a – 8 Multiply the resulting trinomial by the last binomial in the original expression. (a + 3)(15a2 – 14a – 8) Multiply the trinomial by the first term in the binomial, a. [a(15a2 – 14a – 8)] Use the distributive property of multiplication. [a(15a2) – a(14a) – a(8)] Simplify terms. [15a3 – 14a2 – 8a] Multiply the trinomial by the second term in the binomial, 3. [3(15a2) – 3(14a) – 3(8)] Use the distributive property of multiplication. [45a2 – 42a – 24] Add the results of multiplying by the terms in the binomial together. [15a3 – 14a2 – 8a] + [45a2 – 42a – 24] Use the commutative property of addition. 15a3 – 14a2 + 45a2 – 8a – 42a – 24 Combine like terms.
 23. Multiply the first two parenthetical terms in the expression using FOIL. Multiply the first terms in each binomial. ([2n] – 3)([2n] + 3) 4n2 Multiply the outer terms in each binomial. ([2n] – 3)(2n + [3]) +6n Multiply the inner terms in each binomial. (2n – [3])([2n] + 3) –6n Multiply the last terms in each binomial. (2n – [3])(2n + [3]) –9 Add the products of FOIL together. 4n2 + 6n – 6n – 9 Combine like terms. 4n2 – 9 Now we again have two binomials. Use FOIL to find the solution. (n + 4)(4n2 – 9) Multiply the first terms in each binomial. ([n] + 4)([4n2] – 9) 4n3 Multiply the outer terms in each binomial. ([n] + 4)(4n2 – [9]) –9n Multiply the inner terms in each binomial. (n + [4])([4n2] – 9) +16n2 Multiply the last terms in each binomial. (n + [4])(4n2 – [9]) –36 Add the products of FOIL together. 4n3 – 9n + 16n2 – 36 Order terms from the highest to lowest power.
 24. Multiply the trinomial by the first term in the binomial, 5r. [5r(3r4 + 2r2 + 6)] Use the distributive property of multiplication. [5r(3r4) + 5r(2r2) + 5r(6)] Simplify terms. [15r5 + 10r3 + 30r] Multiply the trinomial by the second term in the binomial, –7. [–7(3r4 + 2r2 + 6)] Use the distributive property of multiplication. [–7(3r4) – 7(2r2) – 7(6)] Simplify terms. [–21r4 – 14r2 – 42] Add the results of multiplying by the terms in the binomial together. [15r5 + 10r3 + 30r] + [–21r4 – 14r2 – 42] Use the commutative property of addition.
 25. Multiply the first two parenthetical terms in the expression using FOIL. Multiply the first terms in each binomial. ([3x2] + 4)([x] – 3) 3x3 Multiply the outer terms in each binomial. ([3x2] + 4)(x – [3]) –9x2 Multiply the inner terms in each binomial. (3x2 + [4])([x] – 3) +4x Multiply the last terms in each binomial. (3x2 + [4])(x – [3]) –12 Add the products of FOIL together. (3x3 – 9x2 + 4x – 12) Multiply the resulting trinomial by the last binomial in the original expression. (3x2 – 4)(3x3 – 9x2 + 4x – 12) Multiply the trinomial by the first term in the binomial, 3x2. [3x2(3x3 – 9x2 + 4x – 12)] Use the distributive property of multiplication. [3x2 (3x3) – 3x2(9x2) + 3x2(4x) – 3x2(12)] Simplify terms. [9x5 – 27x4 + 12x3 – 36x2] Multiply the trinomial by the second term in the binomial, –4. [–4(3x3 – 9x2 + 4x – 12)] Use the distributive property of multiplication. [–12x3 + 36x2 – 16x + 48] Add the results of multiplying by the terms in the binomial together. [9x5 – 27x4 + 12x3 – 36x2] + [–12x3 + 36x2 – 16x + 48] Use the commutative property of addition. 9x5 – 27x4 + 12x3 – 12x3 – 36x2 + 36x2 – 16x + 48 Combine like terms.

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