➕ Algebra 1 · Exponents and Polynomials

Adding and Subtracting Polynomials practice questions

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Answer key with explanations

1. Which of these is a trinomial?

   Answer: 3x² − 7x + 1. A trinomial has exactly three terms. 3x² − 7x + 1 has three terms. 9x² has one term (a monomial), x⁴ − 12 has two terms (a binomial), and 4x⁵ − x³ + 2x − 1 has four terms.

2. What is the degree of 2x − 7x⁴ + 5x² + 9?

   Answer: 4. With one variable, the degree is the highest exponent. The exponents are 1, 4, 2 and 0, so the highest is 4 (from −7x⁴). The choice 1 is the exponent of the first term written, 9 is the constant, and 7 comes from the coefficient −7, not from an exponent.

3. Add (6x² − x + 4) + (3x² + 9x − 11).

   Answer: 9x² + 8x − 7. Combine like terms. x²: 6 + 3 = 9. x: −1 + 9 = 8. Constants: 4 + (−11) = −7. So the sum is 9x² + 8x − 7. The choice 9x⁴ added the exponents, 10x dropped the minus sign on −x, and +15 dropped the minus sign on −11.

4. Subtract (9x² + 4x − 6) − (5x² − 2x + 3).

   Answer: 4x² + 6x − 9. Flip every sign in the second polynomial: −(5x² − 2x + 3) = −5x² + 2x − 3. Then 9x² + 4x − 6 − 5x² + 2x − 3. The x² terms: 9 − 5 = 4. The x terms: 4 + 2 = 6. The constants: −6 − 3 = −9. Check with x = 3: the original is 87 − 42 = 45, and 4(9) + 6(3) − 9 = 45. The choice 4x² + 2x − 3 flipped only the first sign, 4x² + 6x − 3 did not flip the last sign, and 14x² + 2x − 3 added the polynomials.

5. A rectangle has length (4x + 3) meters and width (2x − 1) meters. Which expression gives its perimeter in meters?

   Answer: 12x + 4. A rectangle has two lengths and two widths, so add all four sides: (4x + 3) + (2x − 1) + (4x + 3) + (2x − 1). The x terms: 4 + 2 + 4 + 2 = 12. The constants: 3 − 1 + 3 − 1 = 4. The perimeter is 12x + 4. Check with x = 3: the length is 15 and the width is 5, so the perimeter is 40, and 12(3) + 4 = 40. The choice 6x + 2 is only half the perimeter, 12x + 8 treats each −1 as +1, and 12x² + 4 changes the exponent, which adding never does.

6. Simplify (3 + 5x²) − (7 − 4x + x²).

   Answer: 4x² + 4x − 4. Flip every sign in the second group: −(7 − 4x + x²) = −7 + 4x − x². So the expression is 3 + 5x² − 7 + 4x − x². The x² terms: 5 − 1 = 4. The x term: +4x. The constants: 3 − 7 = −4. In standard form, 4x² + 4x − 4. Check with x = 3: the original is 48 − 4 = 44, and 4(9) + 12 − 4 = 44. The choice 4x² − 4x − 4 left the −4x unflipped, 4x² + 4x + 10 left the 7 unflipped, and 6x² − 4x + 10 added the polynomials.

7. The perimeter of a triangle is (10x + 6) centimeters. Two sides are (4x + 1) and (3x + 5) centimeters. What is the third side?

   Answer: 3x. The third side is the perimeter minus the other two sides: (10x + 6) − (4x + 1) − (3x + 5). Flip the signs of the subtracted sides: 10x + 6 − 4x − 1 − 3x − 5. The x terms: 10 − 4 − 3 = 3. The constants: 6 − 1 − 5 = 0. So the third side is 3x centimeters. Check by adding the sides: (4x + 1) + (3x + 5) + 3x = 10x + 6. The choice 3x + 12 added the constants instead of subtracting them, 6x + 5 subtracted only one side, and 17x + 12 added all three.

8. Priya simplified (5x² − 2x + 8) − (3x² + x − 4) and got 2x² − 3x + 4. Which statement describes her mistake and gives the correct answer?

   Answer: She did not flip the sign of the last term. −(−4) = +4, so the constant is 8 + 4 = 12 and the correct answer is 2x² − 3x + 12.. Flip every sign in the second polynomial: −(3x² + x − 4) = −3x² − x + 4. Then 5x² − 2x + 8 − 3x² − x + 4. The x² terms: 5 − 3 = 2. The x terms: −2 − 1 = −3. The constants: 8 + 4 = 12. Priya’s x² and x terms are right, but her constant used 8 − 4, as if the −4 had not been flipped. Check with x = 3: the original is 47 − 26 = 21. Priya’s answer gives 18 − 9 + 4 = 13, but 2x² − 3x + 12 gives 18 − 9 + 12 = 21.

9. A patio is made of two rectangular parts. Their areas are (4x² + 2x) and (x² + 6x + 3) square feet. What is the total area in square feet?

   Answer: 5x² + 8x + 3. The total area is the sum of the two areas. The x² terms: 4 + 1 = 5. The x terms: 2 + 6 = 8. The constants: 0 + 3 = 3 (the first part has no constant term). So the total is 5x² + 8x + 3 square feet. Check with x = 3: the parts are 42 and 30, so the total is 72, and 5(9) + 24 + 3 = 72. The choice 13x + 3 treats x² like x, 5x² + 8x drops the constant, and 5x⁴ + 8x² + 3 changes the exponents, which adding never does.

10. What is the degree of the sum (4x³ + 2x) + (−4x³ + 7x² − 1)?

    Answer: 2. Add like terms. The x³ terms cancel: 4x³ + (−4x³) = 0. What is left is 7x² + 2x − 1, whose highest exponent is 2. So the degree is 2, even though both original polynomials have degree 3. The choice 3 forgets that the leading terms cancel, 6 adds the degrees (adding polynomials does not do that), and 7 is a coefficient, not an exponent.