Holt McDougal Algebra 2 Fundamental Theorem of Algebra • How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots? • How do we identify all of the roots of a polynomial equation?
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Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.
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Holt McDougal Algebra 2
Fundamental Theorem of Algebra
• How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots?
• How do we identify all of the roots of a polynomial equation?
Holt McDougal Algebra 2
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 McDougal Algebra 2
Fundamental Theorem of Algebra Example 1: Writing Polynomial Functions
Write the simplest polynomial with roots –1, , and 4.2 3
1 xxP 23 x 4x
1 xxP 3 2x x14 8
33xxP 214x x8 23x x14 8 33xxP 211x x6 8
Holt McDougal Algebra 2
Fundamental Theorem of Algebra Example 2: Writing Polynomial Functions
Write the simplest polynomial with roots –2, 2, and 4.
2x 2 xxP 4x
2 xxP 2x x6 8
3xxP 26x x8 22x x12 16 3xxP 24 x x4 16
Holt McDougal Algebra 2
Fundamental Theorem of Algebra Example 3: Writing Polynomial Functions
Write the simplest polynomial with roots 0, , and 3.2 3
xxP 23 x 3x
xxP 3 2x x11 6
33xxP 211x x6
Holt McDougal Algebra 2
Fundamental Theorem of Algebra Example 4: Writing Polynomial Functions
Write the simplest polynomial with roots –1, 5, and .3
Fundamental Theorem of Algebra Example 5: Writing Polynomial Functions
Write the simplest polynomial with roots 2, 3i, and .5
,3i 5
2x 2 xxP 2x 529i 2 xxP 4x 24x 45
ix 3 2 xxP 5x 5x ix 3
5xxP x45 42 x 28x 9034x 5xxP 42 x 28x x45 9034x
Holt McDougal Algebra 2
Fundamental Theorem of Algebra Example 6: Writing Polynomial Functions
Write the simplest polynomial with roots 1, 2 + i, and .3
,2 i 3
ix 2 1 xxP 3x 3x ix 2 2x 1 xxP 2x 3x2 ix x2 4 i2 ix i2 2i 1
1 xxP 2x x4 5 32 x 3xxP 2x x3 3 542 xx 5xxP 44x x15212x33x25x34x4x35x
15x1223x 5xxP 45x 36x 210x x27 15
Holt McDougal Algebra 2
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.