Algebra 1 Adding, Subtracting and Multiplying Polynomials Worksheet Solutions (PDF)

Algebra 1 math worksheet solutions: Adding, Subtracting and Multiplying Polynomials – Maths PDF

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This algebra 1 polynomial operations answers for Algebra 1 covers adding, subtracting and multiplying polynomials (Common Core A-APR.A.1), with a free PDF to print.

Algebra 1 polynomial operations answers (Algebra 1) – first page of the printable PDF

Here are full solutions for the polynomial operations worksheet. Read each step, not just the bold answer, and compare it with your own method. Redo any problem you missed, and note which kind of mistake you made so you can avoid it on the test.

Algebra 1 · Common Core State Standards: A-APR.A.1

Algebra 1 polynomial operations answers: exercises 1 to 10

Exercise 1 – Name and degree

It has three terms and the greatest exponent is 3.

Trinomial of degree 3

Exercise 2 – Standard form

Order the terms by descending exponent.

$-x^{3} + 3x^{2} + 4x + 2$. The leading term is $-x^{3}$.

$-x^{3} + 3x^{2} + 4x + 2$; leading coefficient $-1$

Exercise 3 – Add polynomials

Combining like terms.

Combine like terms: $(3 + 1)x^{2} + (2 – 4)x + (-5 + 9)$.

$4x^{2} – 2x + 4$

Exercise 4 – Subtract polynomials

Distribute the negative: $5x^{2} + 3x – 2 – 2x^{2} + x – 6$.

Combine: $3x^{2} + 4x – 8$.

$3x^{2} + 4x – 8$

Exercise 5 – Two variables

Combine the a terms and the b terms: $6a – 2b$.

$6a – 2b$

Exercise 6 – Monomial times polynomial

Multiply 3x by each term: $6x^{3}$, $-15x^{2}$, $3x$.

$6x^{3} – 15x^{2} + 3x$

Exercise 7 – Binomial product

Partial products.

$x^{2} + 7x + 4x + 28$.

$x^{2} + 11x + 28$

Exercise 8 – Binomial with a negative

$x^{2} + 5x – 3x – 15$.

Combine: $x^{2} + 2x – 15$.

$x^{2} + 2x – 15$

Exercise 9 – FOIL

First $6x^{2}$, Outer $-8x$, Inner $3x$, Last $-4$.

Combine $-8x + 3x = -5x$.

$6x^{2} – 5x – 4$

Exercise 10 – Square of a binomial

Area model of the square.

$(x + 6)(x + 6) = x^{2} + 6x + 6x + 36$.

$x^{2} + 12x + 36$

Exercises 11 and up

Exercise 11 – Square with subtraction

$x^{2} + 2(x)(-5) + (-5)^{2}$.

$x^{2} – 10x + 25$

Exercise 12 – Difference of squares

The middle terms cancel: $x^{2} – 8x + 8x – 64$.

$x^{2} – 64$

Exercise 13 – Binomial times trinomial

Distribute x: $x^{3} – 3x^{2} + 4x$. Distribute 2: $2x^{2} – 6x + 8$.

Combine: $x^{3} – x^{2} – 2x + 8$.

$x^{3} – x^{2} – 2x + 8$

Exercise 14 – Closure

$(x^{2} + 1) + (-x^{2} + 3x) = 3x + 1$, which is a polynomial of degree 1.

Polynomials are closed under addition: the sum of two polynomials is always a polynomial, even if the degree drops.

$3x + 1$

Exercise 15 – Check by substitution

The product as an area.

Left side: $(4 + 3)(4 + 2) = 7 \cdot 6 = 42$. Right side: $16 + 20 + 6 = 42$.

Both sides equal 42

Exercise 16 – Rectangle area

$(x + 8)(x + 3) = x^{2} + 3x + 8x + 24 = x^{2} + 11x + 24$.

For $x = 5$: $25 + 55 + 24 = 104$.

$x^{2} + 11x + 24$; 104 square feet

Exercise 17 – Square patio

$(x + 4)^{2} = x^{2} + 8x + 16$.

For $x = 6$: $36 + 48 + 16 = 100$.

$x^{2} + 8x + 16$; 100 square feet

Exercise 18 – Profit polynomial

$P = (-2x^{2} + 60x) – (10x + 100)$.

Distribute the negative: $-2x^{2} + 60x – 10x – 100 = -2x^{2} + 50x – 100$.

For $x = 10$: $-200 + 500 – 100 = 200$.

$-2x^{2} + 50x – 100$; $200

Exercise 19 – Triangle perimeter

$(2x + 3) + (x + 7) + (3x – 2) = 6x + 8$.

For $x = 4$: sides 11, 11, 10, and $6(4) + 8 = 32$.

$6x + 8$; 32 inches

Exercise 20 – Box volume

Volume = $2(x + 4)(x + 1)$. First $(x + 4)(x + 1) = x^{2} + 5x + 4$.

Multiply by 2: $2x^{2} + 10x + 8$. For $x = 3$: $18 + 30 + 8 = 56$.

$2x^{2} + 10x + 8$; 56 cubic inches

Exercise 21 – Deck around a pool

Deck = whole – pool: $(x + 6)(x + 10) – x(x + 4)$.

$(x^{2} + 16x + 60) – (x^{2} + 4x) = 12x + 60$.

For $x = 5$: $12(5) + 60 = 120$.

$12x + 60$; 120 square feet

Key facts behind this algebra 1 polynomial operations answers

Key facts

The degree of a polynomial is its greatest exponent.
Add or subtract by combining like terms; subtracting changes the sign of every term.
Multiply by distributing each term, then combine like terms.
(a + b)² = a² + 2ab + b² and (a + b)(a – b) = a² – b².
The sum, difference, and product of polynomials is a polynomial.

How to use this algebra 1 polynomial operations answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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Algebra 1 Adding, Subtracting and Multiplying Polynomials Worksheet Solutions (PDF)

Algebra 1 polynomial operations answers: exercises 1 to 10

Exercises 11 and up

Key facts behind this algebra 1 polynomial operations answers

How to use this algebra 1 polynomial operations answers

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