If `5x^3+5x^2-6x+9` is divided by (x+3) then the remainder is:

Options

A. 0
B. -32
C. 23
D. -63

Answer

To find the remainder when the polynomial \(5x^3 + 5x^2 - 6x + 9\) is divided by \(x + 3\), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We need to divide the polynomial \(5x^3 + 5x^2 - 6x + 9\) (the dividend) by \(x + 3\) (the divisor).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(5x^3\) by the leading term of the divisor \(x\):
\[ \frac{5x^3}{x} = 5x^2 \]
This is the first term of our quotient.

### Step 3: Multiply and subtract
Multiply the entire divisor \(x + 3\) by \(5x^2\):
\[ 5x^2(x + 3) = 5x^3 + 15x^2 \]
Now subtract this from the original polynomial:
\[ (5x^3 + 5x^2 - 6x + 9) - (5x^3 + 15x^2) = 5x^2 - 15x^2 - 6x + 9 = -10x^2 - 6x + 9 \]

### Step 4: Repeat the process
Now, divide the leading term of the new polynomial \(-10x^2\) by the leading term of the divisor \(x\):
\[ \frac{-10x^2}{x} = -10x \]
Multiply the divisor by \(-10x\):
\[ -10x(x + 3) = -10x^2 - 30x \]
Subtract this from the current polynomial:
\[ (-10x^2 - 6x + 9) - (-10x^2 - 30x) = -6x + 30x + 9 = 24x + 9 \]

### Step 5: Continue the process
Next, divide the leading term \(24x\) by the leading term \(x\):
\[ \frac{24x}{x} = 24 \]
Multiply the divisor by \(24\):
\[ 24(x + 3) = 24x + 72 \]
Subtract this from the current polynomial:
\[ (24x + 9) - (24x + 72) = 9 - 72 = -63 \]

### Step 6: Conclusion
The remainder when \(5x^3 + 5x^2 - 6x + 9\) is divided by \(x + 3\) is \(-63\).

Thus, the final answer is:
\[ \text{Remainder} = -63 \]

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