Find all the real zeros from one known zero

About this lesson

Precalculus › Polynomial and rational functions

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Find all the real zeros from one known zero

Warm-up

Three quick questions from earlier lessons.

Discover

p(x)=x3−2x2−9x+18p(x) = x^3 - 2x^2 - 9x + 18 and p(2)=0p(2) = 0. How many zeros can a polynomial of degree 3 have?

  1. At most 3
  2. Only 1, the one we know

Key idea

One known zero makes a cubic much easier.

If p(r)=0p(r) = 0, divide p(x)p(x) by x−rx - r. The quotient q(x)q(x) has degree one less, and p(x)=(x−r) q(x)p(x) = (x - r)\,q(x). The other zeros are the zeros of qq.

Worked example

Find all the zeros of p(x)=x3−2x2−9x+18p(x) = x^3 - 2x^2 - 9x + 18, knowing that p(2)=0p(2) = 0.

p(x)=x2(x−2)−9(x−2)p(x) = x^2(x - 2) - 9(x - 2)
Group terms to see the factor x − 2 (or use long or synthetic division).
p(x)=(x−2)(x2−9)p(x) = (x - 2)(x^2 - 9)
Factor out x − 2.
Zeros: 22, 33 and −3-3
x² = 9 has two solutions.

Your turn, guided

Your turn: p(x)=(x−1)(x2−16)p(x) = (x - 1)(x^2 - 16).

x=1x = 1
From the first factor.
x=4x = 4 and x=−4x = -4
x² = 16: type the negative solution.

Practice ★

  1. Divide p(x)=x3−2x2−9x+18p(x) = x^3 - 2x^2 - 9x + 18 by x−2x - 2. What is the quotient?
  2. Divide p(x)=x3−6x2−1x+6p(x) = x^3 - 6x^2 - 1x + 6 by x−6x - 6. What is the quotient?
  3. Divide p(x)=x3−3x2−16x+48p(x) = x^3 - 3x^2 - 16x + 48 by x−3x - 3. What is the quotient?

Watch out!

Practice ★★

  1. x=4x = 4 is a zero of p(x)=x3−4x2−1x+4p(x) = x^3 - 4x^2 - 1x + 4. Find all the zeros of pp.
  2. x=3x = 3 is a zero of p(x)=x3−3x2−16x+48p(x) = x^3 - 3x^2 - 16x + 48. Find all the zeros of pp.
  3. x=2x = 2 is a zero of p(x)=x3−2x2−1x+2p(x) = x^3 - 2x^2 - 1x + 2. Find all the zeros of pp.
  1. You know that 22 is a zero of a cubic p(x)p(x). Put the steps for finding all its zeros in order.
    • Check that p(2)=0p(2) = 0
    • Divide p(x)p(x) by x−2x - 2
    • Solve the quadratic quotient =0= 0
    • List 22 with the solutions of the quadratic

Exit ticket

Three questions to check you have got it.

  1. x=4x = 4 is a zero of p(x)=x3−4x2−1x+4p(x) = x^3 - 4x^2 - 1x + 4. Find all the zeros of pp.
  2. x=3x = 3 is a zero of p(x)=x3−3x2−16x+48p(x) = x^3 - 3x^2 - 16x + 48. Find all the zeros of pp.
  3. x=2x = 2 is a zero of p(x)=x3−2x2−1x+2p(x) = x^3 - 2x^2 - 1x + 2. Find all the zeros of pp.

Summary card

All zeros from one zero

  1. Check p(r)=0p(r) = 0.
  2. Divide by x−rx - r: p(x)=(x−r) q(x)p(x) = (x - r)\,q(x).
  3. Solve q(x)=0q(x) = 0; list rr with those solutions.

Watch out: x2=9x^2 = 9 gives 33 and −3-3.

End of the lesson

Next lesson: Complex zeros and conjugate pairs Back to the chapter

Answers: Find all the real zeros from one known zero

A

  1. x2−9x^{2} - 9
  2. x2−1x^{2} - 1
  3. x2−16x^{2} - 16

B

  1. 4, 1, −14,\ 1,\ -1
  2. 3, 4, −43,\ 4,\ -4
  3. 2, 1, −12,\ 1,\ -1

C

  1. 1. Check that p(2)=0p(2) = 0 2. Divide p(x)p(x) by x−2x - 2 3. Solve the quadratic quotient =0= 0 4. List 22 with the solutions of the quadratic

D

  1. 4, 1, −14,\ 1,\ -1
  2. 3, 4, −43,\ 4,\ -4
  3. 2, 1, −12,\ 1,\ -1

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