Algebra 2 Polynomial Operations and Identities Worksheet Solutions (PDF)
This algebra 2 polynomial operations answers for Algebra 2 covers polynomial operations and identities (Common Core A-APR.A.1, A-APR.C.4), with a free PDF to print.
Every worksheet problem on polynomial operations and identities is solved here, one step at a time. Compare your method as well as your final answer, because the steps show where forgetting the middle term of a squared binomial can go wrong. Final answers appear in bold so you can find them quickly when you only need a check.
Algebra 2 · Common Core State Standards: A-APR.A.1, A-APR.C.4
Algebra 2 polynomial operations answers: exercises 1 to 10
Exercise 1 – Add polynomials
Combine like terms: \((3 + 1)x^{2} + (2 – 4)x + (-5 + 9)\).
\(4x^{2} – 2x + 4\)
Exercise 2 – Subtract polynomials
Change every sign in the second polynomial: \(5x^{3} – x + 2 – 2x^{3} – 3x^{2} + 7\).
Combine like terms.
\(3x^{3} – 3x^{2} – x + 9\)
Exercise 3 – Degree and leading coefficient
Standard form lists powers from highest to lowest: \(-3x^{4} + 7x^{2} + 5x – 1\).
Degree 4, leading coefficient -3
Exercise 4 – Monomial times polynomial
\(3x \cdot 2x^{2} = 6x^{3}\), \(3x \cdot (-x) = -3x^{2}\), \(3x \cdot 4 = 12x\).
\(6x^{3} – 3x^{2} + 12x\)
Exercise 5 – Two binomials
\(x^{2} – 7x + 4x – 28\).
\(x^{2} – 3x – 28\)
Exercise 6 – Binomials with coefficients
\(6x^{2} + 10x – 9x – 15\).
\(6x^{2} + x – 15\)
Exercise 7 – Binomial times trinomial
\(x^{3} – 3x^{2} + 4x + 2x^{2} – 6x + 8\).
\(x^{3} – x^{2} – 2x + 8\)
Exercise 8 – Square of a difference
\((a – b)^{2} = a^{2} – 2ab + b^{2}\) with \(a = x\), \(b = 5\).
\(x^{2} – 10x + 25\)
Exercise 9 – Difference of squares
\((3x)^{2} – 2^{2}\).
\(9x^{2} – 4\)
Exercise 10 – Two variables
\((2x)^{2} + 2(2x)(3y) + (3y)^{2}\).
\(4x^{2} + 12xy + 9y^{2}\)
Exercises 11 and up
Exercise 11 – Cube of a binomial
\(x^{3} + 3x^{2}(1) + 3x(1)^{2} + 1^{3}\).
\(x^{3} + 3x^{2} + 3x + 1\)
Exercise 12 – Mental math: a square
\(47^{2} = (50 – 3)^{2} = 2500 – 300 + 9\).
\(2,209\)
Exercise 13 – Mental math: product
\(53 \cdot 47 = (50 + 3)(50 – 3) = 50^{2} – 3^{2} = 2500 – 9\).
\(2,491\)
Exercise 14 – Area of a square
\((x + 5)^{2} = x^{2} + 2(5)x + 25\).
\(x^{2} + 10x + 25\) square feet
Exercise 15 – Verify an identity
\((x + y)^{2} = x^{2} + 2xy + y^{2}\) and \((x – y)^{2} = x^{2} – 2xy + y^{2}\).
Subtract: \(x^{2} + 2xy + y^{2} – x^{2} + 2xy – y^{2} = 4xy\).
Verified: both sides equal \(4xy\)
Exercise 16 – Pythagorean triple
\(x^{2} – y^{2} = 16 – 1 = 15\), \(2xy = 8\), \(x^{2} + y^{2} = 17\).
\(15^{2} + 8^{2} = 225 + 64 = 289 = 17^{2}\).
15, 8, 17
Exercise 17 – Garden path
Outer rectangle: \((x + 6)(x + 10) = x^{2} + 16x + 60\).
Garden: \(x(x + 4) = x^{2} + 4x\).
Subtract: \(x^{2} + 16x + 60 – (x^{2} + 4x)\).
\(12x + 60\) square feet
Exercise 18 – Ticket revenue
\(R = (40 – 2x)(120 + 10x) = 4800 + 400x – 240x – 20x^{2}\).
\(R = -20x^{2} + 160x + 4800\)
Exercise 19 – Difference of cubes of binomials
\((x + 2)^{3} = x^{3} + 6x^{2} + 12x + 8\) and \((x – 2)^{3} = x^{3} – 6x^{2} + 12x – 8\).
Subtract term by term.
\(12x^{2} + 16\)
Exercise 20 – Find the error
Squaring a binomial is not squaring each term. The middle term \(2(x)(6) = 12x\) is missing.
\(x^{2} + 12x + 36\)
Exercise 21 – Prove a pattern
\(2n(2n + 2) + 1 = 4n^{2} + 4n + 1\).
This equals \((2n + 1)^{2}\), a perfect square. Example: \(4 \cdot 6 + 1 = 25 = 5^{2}\).
\((2n + 1)^{2}\)
Key facts behind this algebra 2 polynomial operations answers
- Polynomials are closed under addition, subtraction, and multiplication.
- Combine like terms; subtract by changing every sign.
- Degree of a product = sum of the degrees.
- (a ± b)² = a² ± 2ab + b² and (a + b)(a – b) = a² – b².
- (x² – y²)² + (2xy)² = (x² + y²)² makes Pythagorean triples.
How to use this algebra 2 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.
Download the algebra 2 polynomial operations answers as a PDF with the button below and keep it with your notes.
All Algebra 2 chapters: Algebra 2 math.
More Algebra 2 math
Lessons
See all
Worksheets
See all
Tests
See all
Polynomial Operations and Identities Test with Point Values- Solving Quadratic Equations Test with Point Values
- Radians and the Unit Circle Test with Point Values
- Composition and Inverse Functions Test with Point Values
- Arithmetic and Geometric Sequences and Series Test with Point Values
- Complex Numbers: Adding, Subtracting and Multiplying Test with Point Values
