Given f, of, x, equals, 2, x, cubed, minus, 7, x, plus, kf(x)=2x

Question

3
−7x+k, and x, plus, 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

To find the value of \( k \) given that \( f(x) = 2x^3 - 7x + k \) and \( x + 1 \) is a factor of \( f(x) \), we will use the Factor Theorem.

Factor Theorem: If \( x + 1 \) is a factor of \( f(x) \), then \( f(-1) = 0 \).

Step 1: Evaluate \( f(-1) \)
We substitute \( -1 \) into the function \( f(x) \):

\[ f(-1) = 2(-1)^3 - 7(-1) + k \]

\[ = 2(-1) + 7 + k \]

\[ = -2 + 7 + k \]

\[ = 5 + k \]

Step 2: Set the equation to zero
Since \( f(-1) = 0 \):

\[ 5 + k = 0 \]

Step 3: Solve for \( k \)
Subtract 5 from both sides:

\[ k = -5 \]

Check Answer: Now we will verify that when \( k = -5 \), \( f(x) \) is divisible by \( x + 1 \).

The polynomial is:

\[ f(x) = 2x^3 - 7x - 5 \]

To check if \( (x + 1) \) is a factor, we perform polynomial long division or verify \( f(-1) \):

Substituting \( -1 \):

\[ f(-1) = 2(-1)^3 - 7(-1) - 5 \]

\[ = 2(-1) + 7 - 5 \]

\[ = -2 + 7 - 5 \]

\[ = 0 \]

Since the remainder is 0, \( (x + 1) \) is indeed a factor of \( f(x) \).

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

\[ \boxed{-5} \]