5-8 Rational Zero Theorem
List all of the possible rational zeros of each function.
3
2
11. f (x) = x + x – x – 56
SOLUTION: If
is a rational zero, then p is a factor of –56 and q is a factor of 1.
So, the possible rational zeros are: 6
4
2
13. f (x) = 3x – 4x – x – 35
SOLUTION: If
is a rational zero, then p is a factor of –35 and q is a factor of 3.
So, the possible rational zeros are: 4
3
2
15. f (x) = 8x – 4x – 4x + x + 42
SOLUTION: If
is a rational zero, then p is a factor of 42 and q is a factor of 8.
So, the possible rational zeros are:
4
2
17. f (x) = 16x – 5x + 128
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SOLUTION: If
is a rational zero, then p is a factor of 128 and q is a factor of 16.
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5-8 Rational Zero Theorem
4
2
17. f (x) = 16x – 5x + 128
SOLUTION: If
is a rational zero, then p is a factor of 128 and q is a factor of 16.
So, the possible rational zeros are: Find all of the rational zeros of each function.
3
2
19. f (x) = x + 10x + 31x + 30
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
x = –2 is one of the zeros of the function and the depressed polynomial is
.
Again use the rational root theorem and synthetic division to find the zeros of the depressed polynomial.
The rational zeros are x = –5, –3, –2.
3
2
21. f (x) = 4x – 3x – 100x + 75
SOLUTION: The possible rational zeros are:
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Test for some possible zeros using synthetic division.
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5-8 Rational
Zero
Theorem
The rational
zeros
are x = –5, –3, –2.
3
2
21. f (x) = 4x – 3x – 100x + 75
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
x = 5 is one of the zeros of the function and the depressed polynomial is
.
Again use the rational root theorem and synthetic division to find the zeros of the depressed polynomial.
The rational zeros are
4
3
23. f (x) = x + x – 8x – 8
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
x = –1 is one of the zeros of the function and the depressed polynomial is
.
Again use the rational root theorem and synthetic division to find the zeros of the depressed polynomial.
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The polynomial
doesn’t have rational zeros. Therefore, the rational zeros are
Page 3
5-8 Rational
Zero
Theorem
The rational
zeros
are
4
3
23. f (x) = x + x – 8x – 8
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
x = –1 is one of the zeros of the function and the depressed polynomial is
.
Again use the rational root theorem and synthetic division to find the zeros of the depressed polynomial.
doesn’t have rational zeros. Therefore, the rational zeros are
The polynomial
3
2
25. f (x) = 4x + x + 16x + 4
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is .
Again use the rational root theorem and synthetic division to find the zeros of the depressed polynomial.
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The polynomial
doesn’t have rational zeros. Therefore, the rational zeros of the original polynomial are
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5-8 Rational Zero Theorem
The polynomial
doesn’t have rational zeros. Therefore, the rational zeros are
3
2
25. f (x) = 4x + x + 16x + 4
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is .
Again use the rational root theorem and synthetic division to find the zeros of the depressed polynomial.
The polynomial
doesn’t have rational zeros. Therefore, the rational zeros of the original polynomial are
Find all of the zeros of each function.
3
2
27. f (x) = x + 3x – 25x + 21
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find the zeros.
.
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Therefore, the zeros of the polynomial are x = 1, 3, –7.
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The polynomial
doesn’t have rational zeros. Therefore, the rational zeros of the original polynomial are
5-8 Rational Zero Theorem
Find all of the zeros of each function.
3
2
27. f (x) = x + 3x – 25x + 21
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find the zeros.
.
Therefore, the zeros of the polynomial are x = 1, 3, –7.
4
3
2
29. f (x) = x – x – x – x – 2
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find its zeros.
.
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5-8 Rational
Theorem
Therefore,Zero
the zeros
of the polynomial are x = 1, 3, –7.
4
3
2
29. f (x) = x – x – x – x – 2
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find its zeros.
.
The zeros of the polynomial are
4
3
2
31. f (x) = x – 3x + x – 3x
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find its zeros.
.
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5-8 Rational
Zero Theorem
The zeros of the polynomial are
4
3
2
31. f (x) = x – 3x + x – 3x
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find its zeros.
.
The zeros of the polynomial are
4
3
.
2
33. f (x) = 6x + 22x + 11x – 38x – 40
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find its zeros.
.
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5-8 Rational
Zero Theorem
The zeros of the polynomial are
4
3
.
2
33. f (x) = 6x + 22x + 11x – 38x – 40
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find its zeros.
.
The zeros of the polynomial are
5
3
.
2
35. f (x) = 9x – 94x + 27x + 40x – 12
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find its zeros.
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.
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5-8 Rational
The zerosZero
of theTheorem
polynomial are
5
3
.
2
35. f (x) = 9x – 94x + 27x + 40x – 12
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is Factor the depressed polynomial and find its zeros.
.
The zeros of the polynomial are
4
.
3
37. f (x) = 48x – 52x + 13x – 3
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is eSolutions Manual - Powered by Cognero
Factor the depressed polynomial and find its zeros.
.
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5-8 Rational
The zerosZero
of theTheorem
polynomial are
4
.
3
37. f (x) = 48x – 52x + 13x – 3
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
is one of the zeros of the function and the depressed polynomial is .
Factor the depressed polynomial and find its zeros.
The zeros of the polynomial are
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.
Page 11