Math / Algebra

QuestionIf $x-1$ is a factor of $x^{3}-k x^{2}+2 x$ , what is the value of $k$ ?

Studdy Solution

STEP 1

1. We are given the polynomial $x^3 - kx^2 + 2x$ .
2. We know that $x - 1$ is a factor of this polynomial.
3. We need to find the value of $k$ .

STEP 2

1. Use the Factor Theorem to set up an equation.
2. Substitute the root into the polynomial.
3. Solve the equation for $k$ .

STEP 3

According to the Factor Theorem, if $x - 1$ is a factor, then substituting $x = 1$ into the polynomial should result in zero:
$f(1) = 0$

STEP 4

Substitute $x = 1$ into the polynomial $x^3 - kx^2 + 2x$ :
$f(1) = (1)^3 - k(1)^2 + 2(1)$

STEP 5

Simplify the expression:
$f(1) = 1 - k + 2$ $f(1) = 3 - k$

STEP 6

Set the expression equal to zero and solve for $k$ :
$3 - k = 0$

STEP 7

Solve for $k$ :
$k = 3$
The value of $k$ is:
$\boxed{3}$

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QuestionIf $x-1$ is a factor of $x^{3}-k x^{2}+2 x$ , what is the value of $k$ ?

Studdy Solution

STEP 1

1. We are given the polynomial $x^3 - kx^2 + 2x$ .
2. We know that $x - 1$ is a factor of this polynomial.
3. We need to find the value of $k$ .

STEP 2

1. Use the Factor Theorem to set up an equation.
2. Substitute the root into the polynomial.
3. Solve the equation for $k$ .

STEP 3

According to the Factor Theorem, if $x - 1$ is a factor, then substituting $x = 1$ into the polynomial should result in zero:
$f(1) = 0$

STEP 4

Substitute $x = 1$ into the polynomial $x^3 - kx^2 + 2x$ :
$f(1) = (1)^3 - k(1)^2 + 2(1)$

STEP 5

Simplify the expression:
$f(1) = 1 - k + 2$ $f(1) = 3 - k$

STEP 6

Set the expression equal to zero and solve for $k$ :
$3 - k = 0$

STEP 7

Solve for $k$ :
$k = 3$
The value of $k$ is:
$\boxed{3}$

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QuestionIf x−1x-1x−1 is a factor of x3−kx2+2xx^{3}-k x^{2}+2 xx3−kx2+2x, what is the value of kkk ?

Studdy Solution

STEP 1

1. We are given the polynomial x3−kx2+2x x^3 - kx^2 + 2x x3−kx2+2x.2. We know that x−1 x - 1 x−1 is a factor of this polynomial.3. We need to find the value of k k k.

STEP 2

1. Use the Factor Theorem to set up an equation.2. Substitute the root into the polynomial.3. Solve the equation for k k k.

STEP 3

According to the Factor Theorem, if x−1 x - 1 x−1 is a factor, then substituting x=1 x = 1 x=1 into the polynomial should result in zero:f(1)=0 f(1) = 0 f(1)=0

STEP 4

Substitute x=1 x = 1 x=1 into the polynomial x3−kx2+2x x^3 - kx^2 + 2x x3−kx2+2x:f(1)=(1)3−k(1)2+2(1) f(1) = (1)^3 - k(1)^2 + 2(1) f(1)=(1)3−k(1)2+2(1)

STEP 5

Simplify the expression:f(1)=1−k+2 f(1) = 1 - k + 2 f(1)=1−k+2 f(1)=3−k f(1) = 3 - k f(1)=3−k

STEP 6

Set the expression equal to zero and solve for k k k:3−k=0 3 - k = 0 3−k=0

STEP 7

Solve for k k k:k=3 k = 3 k=3 The value of k k k is:3 \boxed{3} 3​

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QuestionIf $x-1$ is a factor of $x^{3}-k x^{2}+2 x$ , what is the value of $k$ ?

1. We are given the polynomial $x^3 - kx^2 + 2x$ .
2. We know that $x - 1$ is a factor of this polynomial.
3. We need to find the value of $k$ .

1. Use the Factor Theorem to set up an equation.
2. Substitute the root into the polynomial.
3. Solve the equation for $k$ .

According to the Factor Theorem, if $x - 1$ is a factor, then substituting $x = 1$ into the polynomial should result in zero:
$f(1) = 0$

Substitute $x = 1$ into the polynomial $x^3 - kx^2 + 2x$ :
$f(1) = (1)^3 - k(1)^2 + 2(1)$

Simplify the expression:
$f(1) = 1 - k + 2$ $f(1) = 3 - k$

Set the expression equal to zero and solve for $k$ :
$3 - k = 0$

Solve for $k$ :
$k = 3$
The value of $k$ is:
$\boxed{3}$