Math / Algebra

Question.pembuat fungsi nol polinomial $f(x)=4 x^{4}+8 x^{3}-3 x^{2}-7 x-2$ adalah ... A. $x=-2$ atau $x=-1$ atau $x=-1 / 2$ B. $x=-2$ atau $x=1$ atau $x=-1 / 2$ C. $x=2$ atau $x=-1$ atau $x=-1 / 2$ D. $x=2$ atau $x=1$ atau $x=-1 / 2$ E. $x=-2$ atau $x=1$ atau $x=1 / 2$

Studdy Solution

STEP 1

1. We are given the polynomial function $f(x) = 4x^4 + 8x^3 - 3x^2 - 7x - 2$ .
2. We need to find the roots of the polynomial, i.e., the values of $x$ for which $f(x) = 0$ .
3. We have multiple choice options to evaluate.

STEP 2

1. Use the Rational Root Theorem to identify possible rational roots.
2. Test the possible roots using synthetic division or direct substitution.
3. Determine which option contains the correct roots.

STEP 3

Apply the Rational Root Theorem. The possible rational roots are the factors of the constant term $-2$ divided by the factors of the leading coefficient $4$ .
Possible rational roots: $\pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{4}$ .

STEP 4

Test each possible root by substituting into $f(x)$ to see if it equals zero.
Test $x = -2$ :
$f(-2) = 4(-2)^4 + 8(-2)^3 - 3(-2)^2 - 7(-2) - 2$ $f(-2) = 4(16) + 8(-8) - 3(4) + 14 - 2$ $f(-2) = 64 - 64 - 12 + 14 - 2$ $f(-2) = 0$
$x = -2$ is a root.

STEP 5

Test $x = 1$ :
$f(1) = 4(1)^4 + 8(1)^3 - 3(1)^2 - 7(1) - 2$ $f(1) = 4 + 8 - 3 - 7 - 2$ $f(1) = 0$
$x = 1$ is a root.

STEP 6

Test $x = -\frac{1}{2}$ :
$f\left(-\frac{1}{2}\right) = 4\left(-\frac{1}{2}\right)^4 + 8\left(-\frac{1}{2}\right)^3 - 3\left(-\frac{1}{2}\right)^2 - 7\left(-\frac{1}{2}\right) - 2$ $f\left(-\frac{1}{2}\right) = 4\left(\frac{1}{16}\right) + 8\left(-\frac{1}{8}\right) - 3\left(\frac{1}{4}\right) + \frac{7}{2} - 2$ $f\left(-\frac{1}{2}\right) = \frac{1}{4} - 1 - \frac{3}{4} + \frac{7}{2} - 2$ $f\left(-\frac{1}{2}\right) = 0$
$x = -\frac{1}{2}$ is a root.

STEP 7

Based on the tests, the roots are $x = -2$ , $x = 1$ , and $x = -\frac{1}{2}$ .
The correct option is:
B. $x = -2$ or $x = 1$ or $x = -\frac{1}{2}$
The correct answer is option B.

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Question.pembuat fungsi nol polinomial $f(x)=4 x^{4}+8 x^{3}-3 x^{2}-7 x-2$ adalah ... A. $x=-2$ atau $x=-1$ atau $x=-1 / 2$ B. $x=-2$ atau $x=1$ atau $x=-1 / 2$ C. $x=2$ atau $x=-1$ atau $x=-1 / 2$ D. $x=2$ atau $x=1$ atau $x=-1 / 2$ E. $x=-2$ atau $x=1$ atau $x=1 / 2$

Studdy Solution

STEP 1

1. We are given the polynomial function $f(x) = 4x^4 + 8x^3 - 3x^2 - 7x - 2$ .
2. We need to find the roots of the polynomial, i.e., the values of $x$ for which $f(x) = 0$ .
3. We have multiple choice options to evaluate.

STEP 2

1. Use the Rational Root Theorem to identify possible rational roots.
2. Test the possible roots using synthetic division or direct substitution.
3. Determine which option contains the correct roots.

STEP 3

Apply the Rational Root Theorem. The possible rational roots are the factors of the constant term $-2$ divided by the factors of the leading coefficient $4$ .
Possible rational roots: $\pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{4}$ .

STEP 4

Test each possible root by substituting into $f(x)$ to see if it equals zero.
Test $x = -2$ :
$f(-2) = 4(-2)^4 + 8(-2)^3 - 3(-2)^2 - 7(-2) - 2$ $f(-2) = 4(16) + 8(-8) - 3(4) + 14 - 2$ $f(-2) = 64 - 64 - 12 + 14 - 2$ $f(-2) = 0$
$x = -2$ is a root.

STEP 5

Test $x = 1$ :
$f(1) = 4(1)^4 + 8(1)^3 - 3(1)^2 - 7(1) - 2$ $f(1) = 4 + 8 - 3 - 7 - 2$ $f(1) = 0$
$x = 1$ is a root.

STEP 6

STEP 7

Based on the tests, the roots are $x = -2$ , $x = 1$ , and $x = -\frac{1}{2}$ .
The correct option is:
B. $x = -2$ or $x = 1$ or $x = -\frac{1}{2}$
The correct answer is option B.

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Studdy Solution

STEP 1

1. We are given the polynomial function f(x)=4x4+8x3−3x2−7x−2 f(x) = 4x^4 + 8x^3 - 3x^2 - 7x - 2 f(x)=4x4+8x3−3x2−7x−2.2. We need to find the roots of the polynomial, i.e., the values of x x x for which f(x)=0 f(x) = 0 f(x)=0.3. We have multiple choice options to evaluate.

STEP 2

1. Use the Rational Root Theorem to identify possible rational roots.2. Test the possible roots using synthetic division or direct substitution.3. Determine which option contains the correct roots.

STEP 3

Apply the Rational Root Theorem. The possible rational roots are the factors of the constant term −2-2−2 divided by the factors of the leading coefficient 444.Possible rational roots: ±1,±2,±12,±14 \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{4} ±1,±2,±21​,±41​.

STEP 4

STEP 5

Test x=1 x = 1 x=1:f(1)=4(1)4+8(1)3−3(1)2−7(1)−2 f(1) = 4(1)^4 + 8(1)^3 - 3(1)^2 - 7(1) - 2 f(1)=4(1)4+8(1)3−3(1)2−7(1)−2 f(1)=4+8−3−7−2 f(1) = 4 + 8 - 3 - 7 - 2 f(1)=4+8−3−7−2 f(1)=0 f(1) = 0 f(1)=0x=1 x = 1 x=1 is a root.

STEP 6

STEP 7

Based on the tests, the roots are x=−2 x = -2 x=−2, x=1 x = 1 x=1, and x=−12 x = -\frac{1}{2} x=−21​.The correct option is:B. x=−2 x = -2 x=−2 or x=1 x = 1 x=1 or x=−12 x = -\frac{1}{2} x=−21​ The correct answer is option B.

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1. We are given the polynomial function $f(x) = 4x^4 + 8x^3 - 3x^2 - 7x - 2$ .
2. We need to find the roots of the polynomial, i.e., the values of $x$ for which $f(x) = 0$ .
3. We have multiple choice options to evaluate.

1. Use the Rational Root Theorem to identify possible rational roots.
2. Test the possible roots using synthetic division or direct substitution.
3. Determine which option contains the correct roots.

Apply the Rational Root Theorem. The possible rational roots are the factors of the constant term $-2$ divided by the factors of the leading coefficient $4$ .
Possible rational roots: $\pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{4}$ .

Test $x = 1$ :
$f(1) = 4(1)^4 + 8(1)^3 - 3(1)^2 - 7(1) - 2$ $f(1) = 4 + 8 - 3 - 7 - 2$ $f(1) = 0$
$x = 1$ is a root.

Based on the tests, the roots are $x = -2$ , $x = 1$ , and $x = -\frac{1}{2}$ .
The correct option is:
B. $x = -2$ or $x = 1$ or $x = -\frac{1}{2}$
The correct answer is option B.