Zeros of a polynomial and its factors, in both directions

Answer Key - Grade 11 Math Worksheet

Circle the best answer, or write your answer on the line.

1. 1.

   Kira's robot graphs p(x) = (3x + 4)(x − 2)³. Which zero of p comes from the factor 3x + 4?

   -4/3
2. 2.

   Luz lists the zeros of p(x) = (4x − 5)x(x + 1)². Which zero of p comes from the factor 4x − 5?

   5/4
3. 3.

   Milo marks p(x) = (5x + 2)(x − 4)² on a grid map. Which zero of p comes from the factor 5x + 2?

   -2/5
4. 4.

   Noah designs a puzzle: a polynomial with leading coefficient 1 and only the zeros -2 (multiplicity 2), 3 and 5. Which polynomial is it?

   • (x + 2)(x − 3)(x − 5)
   • 2(x + 2)(x − 3)(x − 5)
   • (x + 2)²(x − 3)(x − 5)
   • (x − 2)²(x + 3)(x + 5)
5. 5.

   Ren's robot needs a polynomial with leading coefficient 1 and only the zeros 4 (multiplicity 3) and -1. Which polynomial is it?

   • 3(x − 4)(x + 1)
   • (x − 4)(x + 1)
   • (x + 4)³(x − 1)
   • (x − 4)³(x + 1)
6. 6.

   Zee writes a math club card: leading coefficient 1, only the zeros -5 (multiplicity 2), 1 and 6. Which polynomial is it?

   • (x − 5)²(x + 1)(x + 6)
   • 2(x + 5)(x − 1)(x − 6)
   • (x + 5)²(x − 1)(x − 6)
   • (x + 5)(x − 1)(x − 6)
7. 7.

   Bo knows x = 3 is a zero of p(x) = x³ − 2x² − 5x + 6. What is p(x) ÷ (x − 3)?

   x² + x − 2
8. 8.

   Esme's robot finds that x = -4 is a zero of p(x) = x³ − 3x² − 18x + 40. What is p(x) ÷ (x + 4)?

   x² − 7x + 10
9. 9.

   Jun checks that x = 2 is a zero of p(x) = x³ − 2x² + 3x − 6. What is p(x) ÷ (x − 2)?

   x² + 3
10. 10.

    Kira reads that x = 3 is a zero of p(x) = x⁴ − 10x² + 9. What is p(x) ÷ (x − 3)?

    x³ + 3x² − x − 3