Question
decompose thew polynomial x^3+3x^2-4x-12 into its linear factor given x - 2 is a factor
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 perform polynomial long division to divide the polynomial by \( x - 2 \).
### Step 1: Polynomial Long Division
1. **Set up the division**: Divide \( x^3 + 3x^2 - 4x - 12 \) by \( x - 2 \).
2. **Divide the leading terms**: The leading term of the dividend \( x^3 \) divided by the leading term of the divisor \( x \) gives \( x^2 \).
3. **Multiply**: Multiply \( x^2 \) by \( x - 2 \):
\[
x^2(x - 2) = x^3 - 2x^2
\]
4. **Subtract**: Subtract \( (x^3 - 2x^2) \) from \( (x^3 + 3x^2) \):
\[
(x^3 + 3x^2) - (x^3 - 2x^2) = 5x^2
\]
Now bring down the next term \(-4x\):
\[
5x^2 - 4x
\]
5. **Repeat the process**: Divide \( 5x^2 \) by \( x \) which gives \( 5x \).
6. **Multiply**: Multiply \( 5x \) by \( x - 2 \):
\[
5x(x - 2) = 5x^2 - 10x
\]
7. **Subtract**: Subtract \( (5x^2 - 10x) \) from \( (5x^2 - 4x) \):
\[
(5x^2 - 4x) - (5x^2 - 10x) = 6x
\]
Now bring down \(-12\):
\[
6x - 12
\]
8. **Repeat again**: Divide \( 6x \) by \( x \) which gives \( 6 \).
9. **Multiply**: Multiply \( 6 \) by \( x - 2 \):
\[
6(x - 2) = 6x - 12
\]
10. **Subtract**: Subtract \( (6x - 12) \) from \( (6x - 12) \):
\[
(6x - 12) - (6x - 12) = 0
\]
The result is a quotient of \( x^2 + 5x + 6 \) and a remainder of \( 0 \).
### Step 2: Factor the Quotient
Now we need to factor the quadratic \( x^2 + 5x + 6 \). We can do this by finding two numbers that multiply to \( 6 \) and add to \( 5 \). The numbers \( 2 \) and \( 3 \) satisfy this, so we can factor as follows:
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]
### Step 3: Combine Factors
Now, we can write the complete factorization of the original polynomial:
\[
x^3 + 3x^2 - 4x - 12 = (x - 2)(x + 2)(x + 3)
\]
Thus, the linear factors of the polynomial are \( x - 2 \), \( x + 2 \), and \( x + 3 \).
### Step 1: Polynomial Long Division
1. **Set up the division**: Divide \( x^3 + 3x^2 - 4x - 12 \) by \( x - 2 \).
2. **Divide the leading terms**: The leading term of the dividend \( x^3 \) divided by the leading term of the divisor \( x \) gives \( x^2 \).
3. **Multiply**: Multiply \( x^2 \) by \( x - 2 \):
\[
x^2(x - 2) = x^3 - 2x^2
\]
4. **Subtract**: Subtract \( (x^3 - 2x^2) \) from \( (x^3 + 3x^2) \):
\[
(x^3 + 3x^2) - (x^3 - 2x^2) = 5x^2
\]
Now bring down the next term \(-4x\):
\[
5x^2 - 4x
\]
5. **Repeat the process**: Divide \( 5x^2 \) by \( x \) which gives \( 5x \).
6. **Multiply**: Multiply \( 5x \) by \( x - 2 \):
\[
5x(x - 2) = 5x^2 - 10x
\]
7. **Subtract**: Subtract \( (5x^2 - 10x) \) from \( (5x^2 - 4x) \):
\[
(5x^2 - 4x) - (5x^2 - 10x) = 6x
\]
Now bring down \(-12\):
\[
6x - 12
\]
8. **Repeat again**: Divide \( 6x \) by \( x \) which gives \( 6 \).
9. **Multiply**: Multiply \( 6 \) by \( x - 2 \):
\[
6(x - 2) = 6x - 12
\]
10. **Subtract**: Subtract \( (6x - 12) \) from \( (6x - 12) \):
\[
(6x - 12) - (6x - 12) = 0
\]
The result is a quotient of \( x^2 + 5x + 6 \) and a remainder of \( 0 \).
### Step 2: Factor the Quotient
Now we need to factor the quadratic \( x^2 + 5x + 6 \). We can do this by finding two numbers that multiply to \( 6 \) and add to \( 5 \). The numbers \( 2 \) and \( 3 \) satisfy this, so we can factor as follows:
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]
### Step 3: Combine Factors
Now, we can write the complete factorization of the original polynomial:
\[
x^3 + 3x^2 - 4x - 12 = (x - 2)(x + 2)(x + 3)
\]
Thus, the linear factors of the polynomial are \( x - 2 \), \( x + 2 \), and \( x + 3 \).