Fundamental Theorem of Algebra

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6-6
Fundamental Theorem of Algebra
Objectives
Use the Fundamental Theorem of Algebra and its
corollary to write a polynomial equation of least
degree with given roots. Identify all of the
roots of a polynomial equation.
Holt Algebra 2
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You have learned several important properties
about real roots of polynomial equations.
You can use this information to write polynomial
function when given in zeros.
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Example 1 Writing Polynomial Functions
Write the simplest polynomial with roots 1, ,
and 4.
If r is a zero of P(x), then x r is a factor of
P(x).
Multiply the first two binomials.
Multiply the trinomial by the binomial.
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Check It Out! Example 1a
Write the simplest polynomial function with the
given zeros.
2, 2, 4
If r is a zero of P(x), then x r is a factor
of P(x).
Multiply the first two binomials.
Multiply the trinomial by the binomial.
P(x) x3 4x2 4x 16
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Check It Out! Example 1b
Write the simplest polynomial function with the
given zeros.
0, , 3
If r is a zero of P(x), then x r is a factor
of P(x).
Multiply the first two binomials.
Multiply the trinomial by the binomial.
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Notice that the degree of the function in Example
1 is the same as the number of zeros. This is
true for all polynomial functions. However, all
of the zeros are not necessarily real zeros.
Polynomials functions, like quadratic functions,
may have complex zeros that are not real numbers.
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Example 2 Finding All Roots of a Polynomial
Solve x4 3x3 5x2 27x 36 0 by finding
all roots.
The polynomial is of degree 4, so there are
exactly four roots for the equation.
p 36, and q 1.
Graph y x4 3x3 5x2 27x 36 to find the
real roots.
Find the real roots at or near 1 and 4.
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Example 2 Continued
Test the possible real roots.
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Example 2 Continued
The polynomial factors into (x 1)(x 4)(x2
9) 0.
The solutions are 4, 1, 3i, 3i.
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Check It Out! Example 2
Solve x4 4x3 x2 16x 20 0 by finding all
roots.
Find the real roots at or near 5 and 1.
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Check It Out! Example 2 Continued
Step 2 Graph y x4 4x3 x2 16x 20 to
find the real roots.
Find the real roots at or near 5 and 1.
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Check It Out! Example 2 Continued
The polynomial factors into (x 5)(x 1)(x2
4) 0.
The solutions are 5, 1, 2i, 2i).
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• HOMEWORK-
• Page 449 1-6, 11-19