Question
Decompose the polynomial x^3+3x^2−4x−12 into its linear factors, given that x−2 is a factor.(1 point)%0D%0AResponses
Answers
GPT-4o mini
To decompose the polynomial \( x^3 + 3x^2 - 4x - 12 \) into its linear factors, given that \( x - 2 \) is a factor, we can use polynomial long division or synthetic division to divide the polynomial by \( x - 2 \).
Let's start with the polynomial division:
1. **Divide the leading term:** Divide the leading term \( x^3 \) by \( x \) (the leading term of \( x - 2 \)). The result is \( x^2 \).
2. **Multiply:** Multiply \( x^2 \) by \( x - 2 \) to get \( x^3 - 2x^2 \).
3. **Subtract:** Subtract \( (x^3 - 2x^2) \) from \( x^3 + 3x^2 \) to get \( (3x^2 + 2x^2) = 5x^2 \).
4. **Bring Down:** Now, bring down the next term, which is \(-4x\), giving \( 5x^2 - 4x \).
5. **Repeat the division:** Divide \( 5x^2 \) by \( x \) to get \( 5x \).
6. **Multiply:** Multiply \( 5x \) by \( x - 2 \) to get \( 5x^2 - 10x \).
7. **Subtract:** Subtract \( (5x^2 - 10x) \) from \( 5x^2 - 4x \) to get \( ( -4x + 10x ) = 6x \).
8. **Bring down:** Bring down the last term, \(-12\), giving \( 6x - 12 \).
9. **Repeat the division:** Divide \( 6x \) by \( x \) to get \( 6 \).
10. **Multiply:** Multiply \( 6 \) by \( x - 2 \) to get \( 6x - 12 \).
11. **Subtract:** Subtract \( 6x - 12 \) from \( 6x - 12 \) to get \( 0 \).
The quotient from the division is:
\[
x^2 + 5x + 6
\]
Next, we need to factor \( x^2 + 5x + 6 \) further:
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]
Now we combine everything:
\[
x^3 + 3x^2 - 4x - 12 = (x - 2)(x + 2)(x + 3)
\]
So the polynomial \( x^3 + 3x^2 - 4x - 12 \) can be decomposed into its linear factors as:
\[
(x - 2)(x + 2)(x + 3)
\]
Let's start with the polynomial division:
1. **Divide the leading term:** Divide the leading term \( x^3 \) by \( x \) (the leading term of \( x - 2 \)). The result is \( x^2 \).
2. **Multiply:** Multiply \( x^2 \) by \( x - 2 \) to get \( x^3 - 2x^2 \).
3. **Subtract:** Subtract \( (x^3 - 2x^2) \) from \( x^3 + 3x^2 \) to get \( (3x^2 + 2x^2) = 5x^2 \).
4. **Bring Down:** Now, bring down the next term, which is \(-4x\), giving \( 5x^2 - 4x \).
5. **Repeat the division:** Divide \( 5x^2 \) by \( x \) to get \( 5x \).
6. **Multiply:** Multiply \( 5x \) by \( x - 2 \) to get \( 5x^2 - 10x \).
7. **Subtract:** Subtract \( (5x^2 - 10x) \) from \( 5x^2 - 4x \) to get \( ( -4x + 10x ) = 6x \).
8. **Bring down:** Bring down the last term, \(-12\), giving \( 6x - 12 \).
9. **Repeat the division:** Divide \( 6x \) by \( x \) to get \( 6 \).
10. **Multiply:** Multiply \( 6 \) by \( x - 2 \) to get \( 6x - 12 \).
11. **Subtract:** Subtract \( 6x - 12 \) from \( 6x - 12 \) to get \( 0 \).
The quotient from the division is:
\[
x^2 + 5x + 6
\]
Next, we need to factor \( x^2 + 5x + 6 \) further:
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]
Now we combine everything:
\[
x^3 + 3x^2 - 4x - 12 = (x - 2)(x + 2)(x + 3)
\]
So the polynomial \( x^3 + 3x^2 - 4x - 12 \) can be decomposed into its linear factors as:
\[
(x - 2)(x + 2)(x + 3)
\]