Question

If f, of, x, equals, 4, x, cubed, plus, 5, x, squared, plus, 3f(x)=4x
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:
\[
\boxed{-681}
\]