Holt Algebra 2

6-6 Fundamental Theorem of Algebra 6-6 Fundamental Theorem of Algebra

Holt Algebra 2

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.

Objectives

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

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.

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

Write the simplest polynomial with roots –1, , and 4.

Example 1: Writing Polynomial Functions

If r is a zero of P(x), then x – r is a factor of P(x).

2 3

P(x) = x3 – x2 – 2x + 8 311

3

Multiply the first two binomials.

Multiply the trinomial by the binomial.

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

Write the simplest polynomial function with the given zeros.

If r is a zero of P(x), then x – r is a factor of P(x).

P(x) = x3– 4x2– 4x + 16

Multiply the first two binomials.

Multiply the trinomial by the binomial.

Check It Out! Example 1a

–2, 2, 4

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

Write the simplest polynomial function with the given zeros.

If r is a zero of P(x), then x – r is a factor of P(x).

P(x) = x3– x2+ 2x11 3

Multiply the first two binomials.

Multiply the trinomial by the binomial.

Check It Out! Example 1b

0, , 32 3

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

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.

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

Example 2: Finding All Roots of a Polynomial

p = –36, and q = 1.

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.

Find the real roots at or near –1 and 4.

Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots.

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

Example 2 Continued

Test the possible real roots.

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

Example 2 Continued

The polynomial factors into (x + 1)(x – 4)(x2 + 9) = 0.

The solutions are 4, –1, 3i, –3i.

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

Solve x4 + 4x3 – x2 +16x – 20 = 0 by finding all roots.

Check It Out! Example 2

Find the real roots at or near –5 and 1.

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

Find the real roots at or near –5 and 1.

Step 2 Graph y = x4 + 4x3 – x2 + 16x – 20 to find the real roots.

Check It Out! Example 2 Continued

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

The polynomial factors into (x + 5)(x – 1)(x2 + 4) = 0.

The solutions are –5, 1, –2i, +2i).

Check It Out! Example 2 Continued

Holt Algebra 2

6-6 Fundamental Theorem of Algebra

HOMEWORK-

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