Zeros of Polynomial Functions
Advanced MathSection 3.4
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Number of zeros
• Any nth degree polynomial can have at most n real zeros
• Using complex numbers, every nth degree polynomial has precisely n zeros (real or imaginary)
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Fundamental Theorem of Algebra
• If f(x) is a polynomial of degree n, where n > 0,
• then f has at least one zero in the complex number system
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Linear Factorization Theorem
• If f(x) is a polynomial of degree n, where n > 0,
• then f has precisely n linear factors
1 2n nf x a x c x c x c
1 2where , , are complex numbersnc c c
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Linear Factorization Theorem applied
1st degree: 5 has exactly one zero -5f x x x
22nd degree: 10 25 has exactly two zeros
5 5 5 and 5
(multiplicity counts: 5 is a repeated zero)
f x x x
f x x x x x
3
2
3rd degree: +9x has exactly three zeros
9 3 3 0, 3
f x x
f x x x x x i x i x x i
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Example
• Find all zeros4 1x
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Rational Zero Test
1 21 2 1 0If the polynomial
has coefficients, every rational zero of has the form
rational zero
where and have no common factors other than 1, and a factor of the
n nn nf x a x a x a x a x a
integer fpq
p qp
0 constant term a factor of the leading coefficient n
aq a
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Using the rational zero test
• List all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficientfactors of constant termpossible rational zeros =
factors of leading coefficent
• Use trial-and-error to determine which, if any are actual zeros of the polynomial
• Can use table on graphing calculator to speed up calculations
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Example
• Use the Rational Zero Test to find the rational zeros
3 24 4 16f x x x x
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Using synthetic division
• Test all factors to see if the remainder is zero
• Can also use graphing calculator to estimate zeros, then only check possibilities near your estimate
3 28 40 525f x x x x
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Examples
• Find all rational zeros
3 22 3 8 3f x x x x
3 22 3 1f x x x
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Conjugate pairs
• If the polynomial has real coefficients,
• then zeros occur in conjugate pairs• If a + bi is a zero, then a – bi also
is a zero.
a bia bi
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Example:
• Find a fourth-degree polynomial function with real coefficients that has zeros -2, -2, and 4i
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Factors of a Polynomial
• Even if you don’t want to use complex numbers
• Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros
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Quadratic factors
• If they can’t be factored farther without using complex numbers, they are irreducible over the reals
2 1x x i x i
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Quadratic factors
• If they can’t be factored farther without using irrational numbers, they are irreducible over the rationals– These are reducible over the reals
2 2 2 2x x x
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Finding zeros of a polynomial function
• If given a complex factor– Its conjugate must be a factor– Multiply the two conjugates – this will give
you a real zero– Use long division or synthetic division to
find more factors• If not given any factors
– Use the rational zero test to find rational zeros
– Factor or use the quadratic formula to find the rest
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Examples
• Use the given zero to find all zeros of the function 3 2 9 9, zero 3f x x x x i
3 24 23 34 10, zero 3f x x x x i
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Examples
• Find all the zeros of the function and write the polynomial as a product of linear factors
2 4 1h x x x
3 26 13 10g x x x x
4 210 9f x x x
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Descartes’s Rule of Signs
– A variation in sign means that two consecutive coefficients have opposite signs
• For a polynomial with real coefficients and a constant term,
• The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer
• The number of negative real zeros is either equal to the variations in sign of f(-x) or less than that number by an even integer.
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Examples
• Determine the possible numbers of positive and negative zeros
3 22 3 1g x x x
3 23 2 3f x x x x
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Upper Bound Rule
– When using synthetic division• If what you try isn’t a factor, but• The number on the outside of the
synthetic division is positive – And each number in the answer is
either positive or zero– then the number on the outside is an
upper bound for the real zeros
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Lower Bound Rule
– When using synthetic division• If what you try isn’t a factor, but• The number on the outside of the
synthetic division is negative – The numbers in the answer are
alternately positive and negative (zeros can count as either)
– then the number on the outside is a lower bound for the real zeros
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Examples
• Use synthetic division to verify the upper and lower bounds of the real zeros
4 34 15
: 4: 1
f x x x
Upper xLower x
3 22 3 12 8
: 4: 3
f x x x x
Upper xLower x
Mathematical Modeling and Variation
Advanced MathSection 3.5
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Two basic types of linear models
• y-intercept is nonzero
• y-intercept is zero
y mx b
y mx
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Direct Variation
• Linear• k is slope• y varies directly as x• y is directly proportional to x
for some nonzero constant ky kx
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Direct Variation as an nth power
• y varies directly as the nth power of x
• y is directly proportional to the nth power of x
for some constant kny kx
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Inverse Variation
• Hyperbola (when k is nonzero)• y varies inversely as x• y is inversely proportional to x
for some constant kkyx
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Inverse Variation as an nth power
• y varies inversely as the nth power of x
• y is inversely proportional to the nth power of x
for some constant kn
kyx
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Joint Variation
• Describes two different direct variations
• z varies jointly as x and y• z is jointly proportional to x and y
for some constant kz kxy
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Joint Variation as an nth and mth power
• z varies jointly as the nth power of x and the mth power of y
• z is jointly proportional to the nth power of x and the mth power of y
y for some constant kn mz kx
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Examples
• Find a math model representing the following statements and find the constants of proportionality
• A varies directly as r2.– When r = 3, A = 9p
• y varies inversely as x– When x = 25, y = 3
• z varies jointly as x and y– When x = 4 and y = 8, z = 64
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