About Finding the Zeros of a Polynomial Function:
When we are asked to find the zeros of a polynomial function, we are trying to find the x-values for which the function is equal to zero. In other words, for which x-values will we have f(x) = 0?
Test Objectives
- Demonstrate an understanding of Descartes' Rule of Signs
- Demonstrate an understanding of the Rational Zeros Theorem
- Demonstrate an understanding of the Remainder Theorem
- Demonstrate an understanding of the Factor Theorem
- Demonstrate an understanding of the Upper and Lower Bounds Theorem
- Demonstrate the ability to find the zeros for a polynomial function
#1:
Instructions: Find all zeros.
a) f(x) = x3 + 5x2 + 8x - 96
b) f(x) = x3 - x2 - 53x + 165
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#2:
Instructions: Find all zeros.
a) f(x) = x3 - 2x2 - 16x + 32
b) f(x) = x4 - 5x3 + 11x2 - 15x
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#3:
Instructions: Find all zeros.
a) f(x) = x5 + 2x4 - 18x3 - 36x2 + 81x + 162
b) f(x) = x4 + 16x3 + 54x2 - 16x - 55
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#4:
Instructions: Find all zeros.
a) f(x) = x4 + 5x3 - 8x - 40
b) f(x) = x4 + 6x3 + 6x2 - 12x - 16
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#5:
Instructions: Find all zeros.
a) f(x) = x4 + 8x3 + 11x2 - 32x - 60
b) f(x) = x5 + x4 - 5x3 + 5x2 + 4x - 6
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}{-}4 \pm 4i, 3$$
$$b)\hspace{.2em}{-}2 \pm \sqrt{37}, 5$$
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#2:
Solutions:
$$a)\hspace{.2em}{\pm} 4, 2$$
$$b)\hspace{.2em}0, 3, 1 \pm 2i$$
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#3:
Solutions:
$$a)\hspace{.2em}{-}2, 3 \hspace{.1em}\text{mult.}\hspace{.1em}2, -3 \hspace{.1em}\text{mult.}\hspace{.1em}2$$
$$b)\hspace{.2em}{\pm}1, -11, -5$$
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#4:
Solutions:
$$a)\hspace{.2em}{-}5, 2, -1 \pm i \sqrt{3}$$
$$b)\hspace{.2em}{-}4, -2, \pm \sqrt{2}$$
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#5:
Solutions:
$$a)\hspace{.2em}{-}5, -3, \pm 2$$
$$b)\hspace{.2em}{\pm}1, -3, 1 \pm i$$
