5-8 Rational Zero Theorem
List all of the possible rational zeros of each function.
3
2
1. f (x) = x – 6x – 8x + 24
SOLUTION: If
is a rational zero, then p is a factor of 24 and q is a factor of 1.
So, the possible rational zeros are: 4
2
2. f (x) = 2x + 3x – x + 15
SOLUTION: If
is a rational zero, then p is a factor of 15 and q is a factor of 2.
So, the possible rational zeros are:
3. CCSS REASONING The volume of the triangular pyramid is 210 cubic inches. Find the dimensions of the solid.
SOLUTION: Use the formula to find the volume of the triangular pyramid.
The volume of the pyramid is:
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SOLUTION: the formula
find the volume of the triangular pyramid.
5-8 Use
Rational
Zero to
Theorem
The volume of the pyramid is:
3
Set the volume of pyramid as 210 cm .
The possible rational zeros are:
There is one change of sign of the coefficients, so by Descartes’ Rule of sign, there is only one positive zero.
Use synthetic division to find the zero.
The only one positive zero of the function is 5. We don’t need to check other zeros as the length cannot be negative.
Therefore, the dimensions are 5 in.× 9 in. × 28 in.
Find all of the rational zeros of each function.
3
2
4. f (x) = x – 6x – 13x + 42
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So, the possible rational zeros are: Page 2
only one
positive
zero of the function is 5. We don’t need to check other zeros as the length cannot be negative.
5-8 The
Rational
Zero
Theorem
Therefore, the dimensions are 5 in.× 9 in. × 28 in.
Find all of the rational zeros of each function.
3
2
4. f (x) = x – 6x – 13x + 42
SOLUTION: So, the possible rational zeros are: Make a table for synthetic division and test possible zeros.
.
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 = –3, 2, 7.
4
3
2
5. f (x) = 2x + 11x + 26x + 29x + 12
SOLUTION: So, the possible rational zeros are:
Make a table for synthetic division and test possible zeros.
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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5-8 The
Rational
Zero
Theorem
rational
zeros
are x = –3, 2, 7.
4
3
2
5. f (x) = 2x + 11x + 26x + 29x + 12
SOLUTION: So, the possible rational zeros are:
Make a table for synthetic division and test possible zeros.
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.
The polynomial
does not have rational zeros. Therefore, the rational zeros are .
Find all of the zeros of each function.
3
2
6. f (x) = 3x – 2x – 8x + 5
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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5-8 The
Rational
Zero Theorem does not have rational zeros. Therefore, the rational zeros are polynomial
.
Find all of the zeros of each function.
3
2
6. f (x) = 3x – 2x – 8x + 5
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
3
.
2
7. f (x) = 8x + 14x + 11x + 3
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
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is one of the zeros of the function and the depressed polynomial is Page 5
.
the zeros
of the polynomial are
5-8 Therefore,
Rational Zero
Theorem
3
.
2
7. f (x) = 8x + 14x + 11x + 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 the zeros.
Therefore, the zeros of the polynomial are
4
3
.
2
8. f (x) = 4x + 13x – 8x + 13x – 12
SOLUTION: The possible rational zeros are:
Test for some possible zeros using synthetic division.
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is one of the zeros of the function and the depressed polynomial is .
5-8 Therefore,
Rational Zero
Theorem
the zeros
of the polynomial are
4
3
.
2
8. f (x) = 4x + 13x – 8x + 13x – 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
2
9. f (x) = 4x – 12x + 25x – 14x – 15
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 The
Rational
Zero
Theorem are
zeros of
the polynomial
4
3
2
9. f (x) = 4x – 12x + 25x – 14x – 15
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 8