If f, of, x, equals, 4, x, cubed, plus, 5, x, squared, plus, 3f(x)=4x

Question

3
+5x
2
+3, then what is the remainder when f, of, xf(x) is divided by x, plus, 6x+6?

Solve the problem just like the example below
⬇️
If
f, of, x, equals, 5, x, squared, minus, 4
f(x)=5x
2
−4, then what is the remainder when
f, of, x
f(x) is divided by
x, minus, 2
x−2?
Use the remainder theorem:
Use the remainder theorem:
The remainder of , f, of, x, divided by, left parenthesis, x, minus, 2, right parenthesis, equals , f, of, 2
The remainder of f(x)÷(x−2) equals f(2)
f, of, 2, equals, 5, left parenthesis, 2, right parenthesis, squared, minus, 4
f(2)=5(2)
2
−4
16
16
The slow way:
The slow way:
x, minus, 2
x−2
5, x, squared
5x
2

minus, 4
−4

2
2
5
5
0
0
minus, 4
−4

down arrow
↓
10
10
20
20

5
5
10
10
16
16

Answer 1

To solve the problem where \( f(x) = 4x^3 + 5x^2 + 3 \), we need to find the remainder when \( f(x) \) is divided by \( x + 6 \).

Step 1: Use the Remainder Theorem

The Remainder Theorem states that the remainder of a polynomial \( f(x) \) when divided by \( x - c \) is equal to \( f(c) \). In this case, since we are dividing by \( x + 6 \), we can rewrite it as \( x - (-6) \).

Step 2: Calculate \( f(-6) \)

We need to evaluate \( f(-6) \): \[ f(-6) = 4(-6)^3 + 5(-6)^2 + 3 \]

Step 2.1: Calculate \( (-6)^3 \)

\[ (-6)^3 = -216 \] Thus, \[ 4(-6)^3 = 4 \times -216 = -864 \]

Step 2.2: Calculate \( (-6)^2 \)

\[ (-6)^2 = 36 \] Therefore, \[ 5(-6)^2 = 5 \times 36 = 180 \]

Step 2.3: Adding it all together

Now, we combine the results: \[ f(-6) = -864 + 180 + 3 \]

Add them step-wise: \[ -864 + 180 = -684 \] Then, \[ -684 + 3 = -681 \]

Conclusion

The remainder when \( f(x) \) is divided by \( x + 6 \) is: \[ \boxed{-681} \]