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Answer :
To find all rational roots of the polynomial [tex]\( h(x) = x^4 - 4x^3 - 70x^2 - 76x - 11 \)[/tex], we'll use the Rational Root Theorem. This theorem states that any rational root, in the form of [tex]\(\frac{p}{q}\)[/tex], is such that [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
For the polynomial [tex]\( h(x) = x^4 - 4x^3 - 70x^2 - 76x - 11 \)[/tex]:
1. Identify the constant term and the leading coefficient:
- The constant term is [tex]\(-11\)[/tex].
- The leading coefficient is [tex]\(1\)[/tex].
2. List the factors of the constant term (-11):
- Factors of [tex]\(-11\)[/tex] are [tex]\(\pm 1, \pm 11\)[/tex].
3. List the factors of the leading coefficient (1):
- Since the leading coefficient is 1, its only factors are [tex]\(\pm 1\)[/tex].
4. Possible rational roots:
- The possible rational roots are the factors of the constant term divided by the factors of the leading coefficient. So, the possible rational roots are:
[tex]\[
\pm 1, \pm 11
\][/tex]
5. Test each possible rational root:
- Test [tex]\( x = 1 \)[/tex]:
[tex]\[
h(1) = 1^4 - 4 \cdot 1^3 - 70 \cdot 1^2 - 76 \cdot 1 - 11 = 1 - 4 - 70 - 76 - 11 = -160
\][/tex]
[tex]\( h(1) \neq 0 \)[/tex]
- Test [tex]\( x = -1 \)[/tex]:
[tex]\[
h(-1) = (-1)^4 - 4 \cdot (-1)^3 - 70 \cdot (-1)^2 - 76 \cdot (-1) - 11
= 1 + 4 - 70 + 76 - 11 = 0
\][/tex]
[tex]\( h(-1) = 0 \)[/tex], so [tex]\( x = -1 \)[/tex] is a rational root.
- Test [tex]\( x = 11 \)[/tex]:
[tex]\[
h(11) = 11^4 - 4 \cdot 11^3 - 70 \cdot 11^2 - 76 \cdot 11 - 11
\][/tex]
The calculations show that [tex]\( h(11) \neq 0 \)[/tex].
- Test [tex]\( x = -11 \)[/tex]:
[tex]\[
h(-11) = (-11)^4 - 4 \cdot (-11)^3 - 70 \cdot (-11)^2 - 76 \cdot (-11) - 11
\][/tex]
The calculations show that [tex]\( h(-11) \neq 0 \)[/tex].
Therefore, the only rational root of the polynomial [tex]\( h(x) \)[/tex] is [tex]\(-1\)[/tex].
So the answer is:
[tex]\[
-1
\][/tex]
For the polynomial [tex]\( h(x) = x^4 - 4x^3 - 70x^2 - 76x - 11 \)[/tex]:
1. Identify the constant term and the leading coefficient:
- The constant term is [tex]\(-11\)[/tex].
- The leading coefficient is [tex]\(1\)[/tex].
2. List the factors of the constant term (-11):
- Factors of [tex]\(-11\)[/tex] are [tex]\(\pm 1, \pm 11\)[/tex].
3. List the factors of the leading coefficient (1):
- Since the leading coefficient is 1, its only factors are [tex]\(\pm 1\)[/tex].
4. Possible rational roots:
- The possible rational roots are the factors of the constant term divided by the factors of the leading coefficient. So, the possible rational roots are:
[tex]\[
\pm 1, \pm 11
\][/tex]
5. Test each possible rational root:
- Test [tex]\( x = 1 \)[/tex]:
[tex]\[
h(1) = 1^4 - 4 \cdot 1^3 - 70 \cdot 1^2 - 76 \cdot 1 - 11 = 1 - 4 - 70 - 76 - 11 = -160
\][/tex]
[tex]\( h(1) \neq 0 \)[/tex]
- Test [tex]\( x = -1 \)[/tex]:
[tex]\[
h(-1) = (-1)^4 - 4 \cdot (-1)^3 - 70 \cdot (-1)^2 - 76 \cdot (-1) - 11
= 1 + 4 - 70 + 76 - 11 = 0
\][/tex]
[tex]\( h(-1) = 0 \)[/tex], so [tex]\( x = -1 \)[/tex] is a rational root.
- Test [tex]\( x = 11 \)[/tex]:
[tex]\[
h(11) = 11^4 - 4 \cdot 11^3 - 70 \cdot 11^2 - 76 \cdot 11 - 11
\][/tex]
The calculations show that [tex]\( h(11) \neq 0 \)[/tex].
- Test [tex]\( x = -11 \)[/tex]:
[tex]\[
h(-11) = (-11)^4 - 4 \cdot (-11)^3 - 70 \cdot (-11)^2 - 76 \cdot (-11) - 11
\][/tex]
The calculations show that [tex]\( h(-11) \neq 0 \)[/tex].
Therefore, the only rational root of the polynomial [tex]\( h(x) \)[/tex] is [tex]\(-1\)[/tex].
So the answer is:
[tex]\[
-1
\][/tex]
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