Find all the roots: the Rational Root Theorem, conjugate pairs and Descartes' Rule of Signs

Grade 11 Math Worksheet - Building Confidence

NameDate

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

1.
Noah studies p(x) = (x − 4)²(x² + 9) for math club. Which describes every root of p, counting repeats?
- three roots: 4, 3i and -3i
- two roots: 4 (two times)
- four roots: 4 (two times), 3i and -3i
- four roots: 4 (two times), 3 and -3, written as real numbers
2.
Ren's robot lists the roots of p(x) = (x + 1)³(x² + 4). Which describes every root of p, counting repeats?
- three roots: -1, 2i and -2i
- five roots: -1 (three times), 2 and -2, with the factor treated as real
- three roots: -1 (three times)
- five roots: -1 (three times), 2i and -2i
3.
Tala writes p(x) = x(x − 2)²(x² + 16) on graph paper. Which describes every root of p, counting repeats?
- five roots: 0, 2 (two times), 4i and -4i
- three roots: 0, 2 (two times)
- five roots: 0, 2 (two times), 4 and -4
- four roots: 0, 2, 4i and -4i
4.
Zee reads p(x) = (x + 5)²(x² + 1) in a notebook. Which describes every root of p, counting repeats?
- three roots: -5, i and -i
- four roots: -5 (two times), i and -i
- four roots: -5 (two times), 1 and -1
- two roots: -5 (two times)
5.
Bo lists the candidates from the Rational Root Theorem for 2x³ − 3x² − 11x + 6. Which list is complete?
- ±1, ±2, ±3, ±6
- ±1, ±2, ±3, ±6, ±1/2, ±3/2
- ±1, ±2, ±1/2, ±1/3, ±2/3, ±1/6
- 1, 2, 3, 6, 1/2, 3/2
6.
Jun lists the candidates from the Rational Root Theorem for 2x³ + 5x² − 4x − 3. Which list is complete?
- 1, 3, 1/2, 3/2
- ±1, ±2, ±1/3, ±2/3
- ±1, ±3, ±1/2, ±3/2
- ±1, ±3
7.
Kira lists the candidates from the Rational Root Theorem for 4x³ − 7x + 3. Which list is complete?
- 1, 3, 1/2, 3/2, 1/4, 3/4
- ±1, ±3, ±1/2, ±3/2, ±1/4, ±3/4
- ±1, ±2, ±4, ±1/3, ±2/3, ±4/3
- ±1, ±3
Luz knows p has real coefficients and 3 − 2i is a root. What other root must p have?
Milo's robot finds that -4 + 5i is a root of a polynomial with real coefficients. What other root must it have?
Noah reads that 1 + 6i is a root of a polynomial with real coefficients. What other root must it have?

Find all the roots: the Rational Root Theorem, conjugate pairs and Descartes' Rule of Signs

Answer Key - Grade 11 Math Worksheet - Building Confidence

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

1.
Noah studies p(x) = (x − 4)²(x² + 9) for math club. Which describes every root of p, counting repeats?
- three roots: 4, 3i and -3i
- two roots: 4 (two times)
- four roots: 4 (two times), 3i and -3i
- four roots: 4 (two times), 3 and -3, written as real numbers
2.
Ren's robot lists the roots of p(x) = (x + 1)³(x² + 4). Which describes every root of p, counting repeats?
- three roots: -1, 2i and -2i
- five roots: -1 (three times), 2 and -2, with the factor treated as real
- three roots: -1 (three times)
- five roots: -1 (three times), 2i and -2i
3.
Tala writes p(x) = x(x − 2)²(x² + 16) on graph paper. Which describes every root of p, counting repeats?
- five roots: 0, 2 (two times), 4i and -4i
- three roots: 0, 2 (two times)
- five roots: 0, 2 (two times), 4 and -4
- four roots: 0, 2, 4i and -4i
4.
Zee reads p(x) = (x + 5)²(x² + 1) in a notebook. Which describes every root of p, counting repeats?
- three roots: -5, i and -i
- four roots: -5 (two times), i and -i
- four roots: -5 (two times), 1 and -1
- two roots: -5 (two times)
5.
Bo lists the candidates from the Rational Root Theorem for 2x³ − 3x² − 11x + 6. Which list is complete?
- ±1, ±2, ±3, ±6
- ±1, ±2, ±3, ±6, ±1/2, ±3/2
- ±1, ±2, ±1/2, ±1/3, ±2/3, ±1/6
- 1, 2, 3, 6, 1/2, 3/2
6.
Jun lists the candidates from the Rational Root Theorem for 2x³ + 5x² − 4x − 3. Which list is complete?
- 1, 3, 1/2, 3/2
- ±1, ±2, ±1/3, ±2/3
- ±1, ±3, ±1/2, ±3/2
- ±1, ±3
7.
Kira lists the candidates from the Rational Root Theorem for 4x³ − 7x + 3. Which list is complete?
- 1, 3, 1/2, 3/2, 1/4, 3/4
- ±1, ±3, ±1/2, ±3/2, ±1/4, ±3/4
- ±1, ±2, ±4, ±1/3, ±2/3, ±4/3
- ±1, ±3
Luz knows p has real coefficients and 3 − 2i is a root. What other root must p have?
3 + 2i
Milo's robot finds that -4 + 5i is a root of a polynomial with real coefficients. What other root must it have?
-4 − 5i
Noah reads that 1 + 6i is a root of a polynomial with real coefficients. What other root must it have?
1 − 6i