Given f, of, x, equals, x, cubed, plus, k, x, minus, 2f(x)=x

Question

3
+kx−2, and x, minus, 1x−1 is a factor of f, of, xf(x), then what is the value of kk?

Solve the problem just like the example below
⬇️
Given
f, of, x, equals, 2, x, cubed, plus, k, x, minus, 7
f(x)=2x
3
+kx−7, and
x, minus, 1
x−1 is a factor of
f, of, x
f(x), then what is the value of
k
k?
Factor Theorem:
Factor Theorem:
left parenthesis, x, minus, 1, right parenthesis, is a factor of , f, of, x, colon
(x−1) is a factor of f(x):
f, of, 1, equals, 0
f(1)=0
2, left parenthesis, 1, right parenthesis, cubed, plus, k, left parenthesis, 1, right parenthesis, minus, 7, equals, 0
2(1)
3
+k(1)−7=0
2, left parenthesis, 1, right parenthesis, plus, k, minus, 7, equals, 0
2(1)+k−7=0
2, plus, k, minus, 7, equals, 0
2+k−7=0
k, minus, 5, equals, 0
k−5=
0
plus, 5, plus, 5
+5=
+5
k, equals, 5
k=
5
Check Answer:
Check Answer:
left parenthesis, 2, x, cubed, plus, 5, x, minus, 7, right parenthesis, divided by, left parenthesis, x, minus, 1, right parenthesis
(2x
3
+5x−7)÷(x−1)
x, minus, 1
x−1
2, x, cubed
2x
3

plus, 5, x
+5x
minus, 7
−7

1
1
2
2
0
0
5
5
minus, 7
−7

down arrow
↓
2
2
2
2
7
7

2
2
2
2
7
7
0
0

When , k, equals, 5, , the remainder is , 0
When k=5, the remainder is 0
So , left parenthesis, x, minus, 1, right parenthesis, is a factor of , f, of, x
So (x−1) is a factor of f(x)

Answer 1

Given \( f(x) = x^3 + kx - 2 \) and \( (x - 1) \) is a factor of \( f(x) \), we need to find the value of \( k \).

Factor Theorem:

According to the Factor Theorem, if \( (x - 1) \) is a factor of \( f(x) \), then:

\[ f(1) = 0 \]

Calculating \( f(1) \):

\[ f(1) = 1^3 + k(1) - 2 = 0 \]

This simplifies to:

\[ 1 + k - 2 = 0 \]

Now, solving for \( k \):

\[ k - 1 = 0 \] \[ k = 1 \]

Check Answer:

To verify, we substitute \( k = 1 \) back into \( f(x) \):

\[ f(x) = x^3 + 1x - 2 = x^3 + x - 2 \]

Now, we should check if \( (x - 1) \) is indeed a factor of \( f(x) \):

We perform polynomial long division of \( f(x) = x^3 + x - 2 \) by \( (x - 1) \):

1. Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( (x - 1) \) to get \( x^3 - x^2 \).
3. Subtract: \( (x^3 + x - 2) - (x^3 - x^2) = x^2 + x - 2 \).
4. Divide \( x^2 \) by \( x \) to get \( x \).
5. Multiply \( x \) by \( (x - 1) \) to get \( x^2 - x \).
6. Subtract: \( (x^2 + x - 2) - (x^2 - x) = 2x - 2 \).
7. Divide \( 2x \) by \( x \) to get \( 2 \).
8. Multiply \( 2 \) by \( (x - 1) \) to get \( 2x - 2 \).
9. Subtract: \( (2x - 2) - (2x - 2) = 0 \).

The remainder is \( 0 \) indicating that \( (x - 1) \) is a factor of \( f(x) \).

Thus, when \( k = 1 \), the remainder is \( 0 \).

So, the value of \( k \) is:

\[ \boxed{1} \]