Question

Simplify: StartFraction x cubed plus 5 x squared minus 20 over x plus 3 EndFraction



Move values to the blanks to complete the equation.

Response area with 4 blank spaces

Blank space 1 empty
Blank space 5 empty
Blank space 7 empty
Blank space 8 empty
Answer options with 7 options.
-6, -2, 1, 2, 8, 24, 52

Answer 1

To simplify the expression \(\frac{x^3 + 5x^2 - 20}{x + 3}\), we can use polynomial long division.

1. **Divide the first term of the numerator by the first term of the denominator**:
- \(x^3 \div x = x^2\)

2. **Multiply the entire divisor by this result**:
- \((x + 3)x^2 = x^3 + 3x^2\)

3. **Subtract this from the original polynomial**:
- \((x^3 + 5x^2 - 20) - (x^3 + 3x^2) = 2x^2 - 20\)

4. **Repeat with the new polynomial**:
- Divide \(2x^2\) by \(x\) to get \(2x\).
- Multiply the entire divisor by \(2x\): \((x + 3)(2x) = 2x^2 + 6x\).
- Subtract: \((2x^2 - 20) - (2x^2 + 6x) = -6x - 20\).

5. **Repeat again**:
- Divide \(-6x\) by \(x\) to get \(-6\).
- Multiply the entire divisor by \(-6\): \((x + 3)(-6) = -6x - 18\).
- Subtract: \((-6x - 20) - (-6x - 18) = -2\).

Now we've completed the polynomial long division. We have:
\[
x^3 + 5x^2 - 20 = (x + 3)(x^2 + 2x - 6) - 2
\]
This means our simplified expression is:
\[
\frac{x^3 + 5x^2 - 20}{x + 3} = x^2 + 2x - 6 - \frac{2}{x + 3}
\]
However, since we're just simplifying to the polynomial part without the remainder:
\[
= x^2 + 2x - 6
\]

Now, to fill in the blanks based on this:
- We can write \(x^2 + 2x - 6\) in the form of \(a + bx + cx^2\):
- \(Blank 1 = -6\)
- \(Blank 5 = 2\)
- \(Blank 7 = 1\) (the coefficient of \(x^2\) is 1)

Thus the completed equation from the options provided is:
- Blank space 1: -6
- Blank space 5: 2
- Blank space 7: 1

The fourth space was not needed based on the simplification path provided, and thus is empty.

Final Answer:
- Blank space 1: -6
- Blank space 5: 2
- Blank space 7: 1
- Blank space 8: [Empty]