Given f, of, x, equals, x, squared, minus, 3, x, plus, kf(x)=x

Question

2
−3x+k, and the remainder when f, of, xf(x) is divided by x, plus, 5x+5 is 5555, 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 solve the problem with \( f(x) = x^2 - 3x + k \) and the given condition about the remainder when divided by \( x + 5 \), we will use the Remainder Theorem.

Remainder Theorem:

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \).

Given:

• \( f(x) = x^2 - 3x + k \)
• We need to find \( k \) such that when \( f(x) \) is divided by \( x + 5 \), the remainder is 5555.

1. Use the Remainder Theorem and find \( f(-5) \): \[ f(-5) = (-5)^2 - 3(-5) + k \]

2. Simplify \( f(-5) \): \[ f(-5) = 25 + 15 + k \] \[ f(-5) = 40 + k \]

3. Set this equal to 5555 since that is the remainder: \[ 40 + k = 5555 \]

4. Solving for \( k \): \[ k = 5555 - 40 \] \[ k = 5515 \]

Conclusion:

The value of \( k \) is \( 5515 \).

Check Answer:

To verify, substitute \( k = 5515 \) back into \( f(-5) \):

1. Calculate again: \[ f(-5) = 40 + 5515 = 5555 \] This confirms that the remainder is indeed 5555 when \( k = 5515 \).

Thus, \( k = 5515 \) is the correct and final answer.