Polynomials Worksheet Answer Key – Algebra 1

Problem 1 Standard Form and Degree

Order the powers from highest to lowest: −4x³ + 7x + 2. The highest power is 3.

−4x³ + 7x + 2; degree 3; leading coefficient −4

Problem 2 Name Each Polynomial

1. One term: monomial, degree 4.
2. Two terms: binomial, degree 2.
3. Three terms: trinomial, degree 3.

(a) monomial, 4; (b) binomial, 2; (c) trinomial, 3

Problem 3 Evaluate a Polynomial

P(−2) = (−2)² − 3(−2) + 5 = 4 + 6 + 5 = 15.

15

Problem 4 Add Two Trinomials

x² terms: 4 + 1 = 5. x terms: 3 − 6 = −3. Constants: −1 + 9 = 8.

5x² − 3x + 8

Problem 5 Subtract Two Trinomials

Distribute the minus sign: 7x² − 2x + 6 − 3x² − 5x + 1.

x²: 7 − 3 = 4. x: −2 − 5 = −7. Constants: 6 + 1 = 7.

4x² − 7x + 7

Problem 6 Multiply Monomials

1. Coefficients: 3 × (−5) = −15. Powers: x² · x⁴ = x⁶. Result −15x⁶.
2. Coefficients: 6 × (−2) = −12. Powers: a³ · a = a⁴ and b · b⁴ = b⁵. Result −12a⁴b⁵.

(a) −15x⁶; (b) −12a⁴b⁵

Problem 7 Distribute a Monomial

4x · 2x² = 8x³. 4x · (−3x) = −12x². 4x · 7 = 28x.

8x³ − 12x² + 28x

Problem 8 Multiply Two Binomials

First: x². Outer: −3x. Inner: 8x. Last: −24.

Combine: −3x + 8x = 5x.

x² + 5x − 24

Problem 9 Binomials with Coefficients

First: 24x². Outer: 54x. Inner: −28x. Last: −63.

Combine: 54x − 28x = 26x.

24x² + 26x − 63

Problem 10 Trinomial Times Binomial

Row x: x³, −4x², 6x. Row 3: 3x², −12x, 18.

Combine: −4x² + 3x² = −x² and 6x − 12x = −6x.

x³ − x² − 6x + 18

Problem 11 Square a Binomial

1. (x)² + 2(x)(7) + 7² = x² + 14x + 49.
2. (3x)² − 2(3x)(2) + 2² = 9x² − 12x + 4.

(a) x² + 14x + 49; (b) 9x² − 12x + 4

Problem 12 Difference of Squares

The outer and inner products cancel: −99k + 99k = 0.

(9k)² − 11² = 81k² − 121.

81k² − 121

Problem 13 Three Polynomials at Once

Remove parentheses: 5x² − 3x + 8 − 2x² − x + 6 + x² − 4.

x²: 5 − 2 + 1 = 4. x: −3 − 1 = −4. Constants: 8 + 6 − 4 = 10.

4x² − 4x + 10

Problem 14 Closure Check

2x · x² = 2x³; 2x · (−3) = −6x; 1 · x² = x²; 1 · (−3) = −3.

Standard form: 2x³ + x² − 6x − 3. It is a polynomial of degree 3, because the degrees add: 1 + 2 = 3.

2x³ + x² − 6x − 3; degree 3

Problem 15 Patio with a Pond

Patio: (x + 9)(x + 4) = x² + 13x + 36. Pond: x².

Difference: x² + 13x + 36 − x² = 13x + 36.

(13x + 36) square feet

Problem 16 Ticket Revenue

Revenue = (40 − 2x)(30 + 5x) = 1200 + 200x − 60x − 10x² = −10x² + 140x + 1200.

At x = 3: −10(9) + 140(3) + 1200 = −90 + 420 + 1200 = 1530.

Check: price 40 − 6 = 34, tickets 30 + 15 = 45, and 34 × 45 = 1,530.

−10x² + 140x + 1200; $1,530

Problem 17 Profit from Revenue and Cost

P(x) = (−3x² + 90x) − (12x + 250) = −3x² + 90x − 12x − 250 = −3x² + 78x − 250.

P(10) = −300 + 780 − 250 = 230.

Check: R(10) = 600, C(10) = 370, and 600 − 370 = 230.

P(x) = −3x² + 78x − 250; $230

Problem 18 Cube of a Binomial

(x + 2)² = x² + 4x + 4.

(x + 2)(x² + 4x + 4) = x³ + 4x² + 4x + 2x² + 8x + 8.

Combine: x³ + 6x² + 12x + 8.

x³ + 6x² + 12x + 8

Problem 19 Arithmetic with Special Products

47 × 53 = (50 − 3)(50 + 3) = 50² − 3² = 2500 − 9 = 2491.

62² = (60 + 2)² = 3600 + 240 + 4 = 3844.

2,491 and 3,844

Problem 20 Find the Error

First: wrong. Squaring means (x − 4)(x − 4), which has the middle term −8x. Correct: x² − 8x + 16.

Second: wrong. The outer and inner products −3x and +3x cancel, so there is no x term. Correct: x² − 9.

(x − 4)² = x² − 8x + 16; (x + 3)(x − 3) = x² − 9