Zeros of polynomials and their multiplicity

About this lesson

Precalculus › Polynomial and rational functions

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Zeros of polynomials and their multiplicity

Warm-up

Three quick questions from earlier lessons.

Discover

p(x)=(x−2)(x+3)p(x) = (x - 2)(x + 3). For which xx is p(x)=0p(x) = 0?

  1. x=2x = 2 and x=−3x = -3
  2. x=2x = 2 and x=3x = 3

Key idea

A product is zero when one of its factors is zero. That gives the zeros of a factored polynomial.

The factor (x−r)m(x - r)^m gives the zero rr with multiplicity mm. If mm is odd, the graph crosses the xx-axis at rr; if mm is even, it touches the axis and turns back.

Worked example

Describe the zeros of p(x)=(x−1)2(x+4)p(x) = (x - 1)^2(x + 4).

x=1x = 1, multiplicity 2: the graph touches the axis
x − 1 = 0, and the factor is squared.
x=−4x = -4, multiplicity 1: the graph crosses
x + 4 = 0, the factor appears once.

Your turn, guided

Your turn: p(x)=(x+5)3(x−2)p(x) = (x + 5)^3(x - 2).

x=2x = 2, multiplicity 1
x − 2 = 0, once: the graph crosses.
x=−5x = -5, multiplicity 3: the graph crosses
x + 5 = 0: type the zero.

Practice ★

  1. Find all the zeros of p(x)=(x−4)(x+2)2p(x) = (x - 4)(x + 2)^2.
  2. Find all the zeros of p(x)=(x−5)(x+1)2p(x) = (x - 5)(x + 1)^2.
  3. Find all the zeros of p(x)=(x−9)(x+1)2p(x) = (x - 9)(x + 1)^2.
  4. Find all the zeros of p(x)=(x−3)(x+1)2p(x) = (x - 3)(x + 1)^2.

Watch out!

Practice ★★

  1. p(x)=(x−4)2(x+9)p(x) = (x - 4)^2(x + 9). What does the graph of pp do at x=4x = 4?
    1. It touches the x-axis and turns back
    2. It crosses the x-axis
    3. It does not reach the x-axis
  2. p(x)=(x−5)2(x+1)p(x) = (x - 5)^2(x + 1). What does the graph of pp do at x=5x = 5?
    1. It touches the x-axis and turns back
    2. It crosses the x-axis
    3. It does not reach the x-axis
  3. p(x)=(x−7)2(x+4)p(x) = (x - 7)^2(x + 4). What does the graph of pp do at x=7x = 7?
    1. It touches the x-axis and turns back
    2. It crosses the x-axis
    3. It does not reach the x-axis
  1. Which numbers are zeros of p(x)=x(x−9)(x+7)p(x) = x(x - 9)(x + 7)? Tick all of them.
    1. 00
    2. 99
    3. −7-7
    4. −9-9
    5. 77
  2. Which numbers are zeros of p(x)=x(x−2)(x+4)p(x) = x(x - 2)(x + 4)? Tick all of them.
    1. 00
    2. 22
    3. −4-4
    4. −2-2
    5. 44

Exit ticket

Three questions to check you have got it.

  1. Find all the zeros of p(x)=(x−4)(x+2)2p(x) = (x - 4)(x + 2)^2.
  2. Find all the zeros of p(x)=(x−5)(x+1)2p(x) = (x - 5)(x + 1)^2.
  3. Find all the zeros of p(x)=(x−9)(x+1)2p(x) = (x - 9)(x + 1)^2.

Summary card

Zeros and multiplicity

  • (x−r)(x - r) gives the zero rr; (x+q)(x + q) gives the zero −q-q.
  • Odd multiplicity: the graph crosses. Even: it touches and turns back.

Watch out: check the sign: x+3=0x + 3 = 0 means x=−3x = -3.

End of the lesson

Next lesson: The remainder theorem and the factor theorem Back to the chapter

Answers: Zeros of polynomials and their multiplicity

A

  1. 4, −24,\ -2
  2. 5, −15,\ -1
  3. 9, −19,\ -1
  4. 3, −13,\ -1

B

  1. It touches the x-axis and turns back
  2. It touches the x-axis and turns back
  3. It touches the x-axis and turns back

C

  1. 00 · 99 · −7-7
  2. 00 · 22 · −4-4

D

  1. 4, −24,\ -2
  2. 5, −15,\ -1
  3. 9, −19,\ -1

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