MATH 1314 - College Algebra
Zeros and x-Intercepts of Polynomials
If p(x) is a polynomial, the solutions to the equation p(x) = 0 are called the zeros of the
polynomial. Sometimes the zeros of a polynomial can be determined by factoring or by using the
Quadratic Formula, but frequently the zeros must be approximated. The real zeros of a polynomial
p(x) are the x-intercepts of the graph of y = p(x) .
--------------------------------------------------------------------The Factor Theorem: If (x − k) is a factor of a polynomial, then x = k is a zero of the polynomial.
Conversely, if x = k is a zero of a polynomial, then (x − k) is a factor of the polynomial.
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-----------------------------------------------Example 1: Find the zeros and x-intercepts of the graph of p(x) =x 4 −5x 2 + 4 .
x 4 −5x 2 + 4 = 0
(x 2 − 4)(x 2 −1) = 0
(x + 2)(x − 2)(x + 1)(x −1) = 0
x + 2 = 0 or x − 2 = 0 or x + 1 = 0 or x −1 = 0
x = −2 or x = 2 or x = −1 or x = 1
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So the zeros are –2, 2, –1, and 1 and the x-intercepts are (–2,0), (2,0), (–1,0), and (1,0).
--------------------------------------------------------------------The number of times a factor occurs in a polynomial is called the multiplicity of the factor. The
corresponding zero is said to have the same multiplicity. For example, if the factor (x − 3) occurs to
the fifth power in a polynomial, then (x − 3) is said to be a factor of multiplicity 5 and the
corresponding zero, x=3, is said to have multiplicity 5. A factor or zero with multiplicity two is
sometimes said to be a double factor or a double zero. Similarly, a factor or zero with multiplicity
three is sometimes said to be a triple factor or a triple zero.
--------------------------------------------------------------------Example 2: Determine the equation, in factored form, of a polynomial p(x) that has 5 as double
zero, –2 as a zero with multiplicity 1, and 0 as a zero with multiplicity 4.
p(x) = (x − 5) 2 (x + 2)x 4
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -€- - - - - - - - - - - - - - - - - - - - - - - - - Example 3: Give the zeros and their multiplicities for p(x) = −12x 4 + 36x 3 − 21x 2 .
−12x 4 + 36x 3 − 21x 2 = 0
−3x 2 (4 x 2 −12x + 7) = 0
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−3x 2 = 0 or 4 x 2 −12x + 7 = 0
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−(−12)± (−12) 2 −4 (4)(7)
2(4)
x2 = 0
or x =
x =0
or x = 12± 144−112
= 12±8 32 = 12±48 2 = 12
± 4 8 2 = 23 ± 22
8
8
So 0 is a zero with multiplicity 2, x = 23 − 22 is a zero with multiplicity 1, and x = 23 + 22 is a zero
with multiplicity 1.
(Thomason - Fall 2008)
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Because the graph of a polynomial is connected, if the polynomial is positive at one value of x and
negative at another value of x, then there must be a zero of the polynomial between those two values
of x.
--------------------------------------------------------------------Example 4: Show that p(x) = 2x 3 − 5x 2 + 4 x − 7 must have a zero between x = 1 and x = 2 .
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p(1) = 2(1) 3 − 5(1) 2 + 4(1) − 7 = 2(1) − 5(1) + 4 − 7 = 2 − 5 + 4 − 7 = −6
and
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p(2) = 2(2) 3 −€
5(2) 2 + 4(2) − 7 = 2(8) − 5(2) + 8 − 7 = 16 −10 + 8 − 7 =€7 .
Because p(1) is negative and p(2) is positive and because the graph of p(x) is connected, p(x)
must equal 0 for a value of x between 1 and 2.
--------------------------------------------------------------------If a factor of a polynomial occurs to an odd power, then the graph of the polynomial actually goes
€across the x-axis at the
€ corresponding x-intercept. An x-intercept
€ of this type is sometimes
€
called an
odd x-intercept. If a factor of a polynomial occurs to an even power, then the graph of the
polynomial "bounces" against the x-axis at the corresponding x-intercept, but not does not go across
the x-axis there. An x-intercept of this type is sometimes called an even x-intercept.
--------------------------------------------------------------------Example 5: Use a graphing calculator or a computer program to graph
y = 0.01x 2 (x + 2) 3 (x − 2)(x − 4) 4 .
Because the factors (x + 2) and (x − 2) appear to odd
y
powers, the graph crosses the x-axis at x = −2
and x = 2 .
5
–2
2
4 x
€
€
€ factors x and
€ (x − 4) appear to even
Because the
€
powers, the graph bounces against the x-axis at x = 0
and x = 4 .
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Note that if the factors
of the polynomial were
€
multipled out, the leading term would be 0.01x10 .
This accounts for the fact that both tails of the graph
go up; in other words, as x → −∞ , y → ∞ and as
x → ∞, y → ∞.
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(Thomason - Fall 2008)
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