LEMMA

Advanced algebra

Zeros and factors of polynomials

ALEKS placement

A root, a factor, and an x-intercept are three names for the same thing.

What this covers

P(c)=0⟺(x−c)dividesP
  • Use the rational root theorem: candidates are ± factors of the constant over factors of the leading coefficient.
  • Test candidates until one gives a remainder of zero.
  • Divide out that factor and repeat on the smaller polynomial.
  • Solve the final quadratic by any method.

Worked example

Worked example

Find the zeros of x³ - 2x² - 5x + 6.

  1. P(1)=1−2−5+6=01 is a candidate and it works, so (x - 1) is a factor.
  2. P(x)=(x−1)(x2−x−6)Divide out (x - 1).
  3. x=1,3,−2x² - x - 6 = (x - 3)(x + 2).
Another worked example

Find the zeros of x³ − 3x + 2 and their multiplicities.

  1. P(1)=1−3+2=01 is a candidate and it works, so x − 1 is a factor.
  2. P(x)=(x−1)(x2+x−2)=(x−1)2(x+2)Divide out x − 1. The leftover quadratic contains x − 1 again.
  3. x∈{−2,1}The factor x − 1 appears twice, so 1 is a zero of multiplicity 2. The zero −2 has multiplicity 1.

A common mistake

A mistake Lemma catches

From (x−1)2(x+2)

x=2x=−2

“A zero is the x that makes a factor 0. For x + 2 that is x = −2, since −2 + 2 = 0. Reading the sign as written gives 2, but 2 + 2 = 4, not 0.”

Try one

Sample problem

One zero of the polynomial is 4. Find the other zero.

P(x)=x2−6x+8

Practice polynomial zeros free

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