Question
What are the linear and quadratic factors of the expression 3x4+7x3+4x2 ?(1 point) Responses (3x+4)(x+1) left parenthesis 3 x plus 4 right parenthesis left parenthesis x plus 1 right parenthesis x2(3x+4)(x+1) x squared left parenthesis 3 x plus 4 right parenthesis left parenthesis x plus 1 right parenthesis x2(3x+4)(x+1)(x+1) x squared left parenthesis 3 x plus 4 right parenthesis left parenthesis x plus 1 right parenthesis left parenthesis x plus 1 right parenthesis x2(3x−4)(x−1)
Answers
GPT-4o mini
To find the linear and quadratic factors of the polynomial \( 3x^4 + 7x^3 + 4x^2 \), we can first factor out the greatest common factor from the expression.
1. **Factoring out the greatest common factor:**
The terms \( 3x^4 \), \( 7x^3 \), and \( 4x^2 \) all have at least \( x^2 \) in common. Therefore, we can factor out \( x^2 \):
\[
3x^4 + 7x^3 + 4x^2 = x^2(3x^2 + 7x + 4)
\]
2. **Factoring the quadratic \( 3x^2 + 7x + 4 \):**
Next, we look to factor the quadratic \( 3x^2 + 7x + 4 \). We want two numbers that multiply to \( 3 \times 4 = 12 \) and add up to \( 7 \). The numbers \( 3 \) and \( 4 \) work:
\[
3x^2 + 3x + 4x + 4 = (3x^2 + 3x) + (4x + 4)
\]
Factoring by grouping gives us:
\[
= 3x(x + 1) + 4(x + 1) = (3x + 4)(x + 1)
\]
3. **Combining factors:**
Now, combine the factors we found:
\[
3x^4 + 7x^3 + 4x^2 = x^2(3x + 4)(x + 1)
\]
So, the final factorization of \( 3x^4 + 7x^3 + 4x^2 \) is:
\[
x^2(3x + 4)(x + 1)
\]
The linear and quadratic factors are \( (3x + 4) \) (linear), \( (x + 1) \) (linear), and \( x^2 \) (which is a quadratic in the sense of having a square term).
Thus, one of the possible responses is:
**\( x^2(3x+4)(x+1) \)**.
1. **Factoring out the greatest common factor:**
The terms \( 3x^4 \), \( 7x^3 \), and \( 4x^2 \) all have at least \( x^2 \) in common. Therefore, we can factor out \( x^2 \):
\[
3x^4 + 7x^3 + 4x^2 = x^2(3x^2 + 7x + 4)
\]
2. **Factoring the quadratic \( 3x^2 + 7x + 4 \):**
Next, we look to factor the quadratic \( 3x^2 + 7x + 4 \). We want two numbers that multiply to \( 3 \times 4 = 12 \) and add up to \( 7 \). The numbers \( 3 \) and \( 4 \) work:
\[
3x^2 + 3x + 4x + 4 = (3x^2 + 3x) + (4x + 4)
\]
Factoring by grouping gives us:
\[
= 3x(x + 1) + 4(x + 1) = (3x + 4)(x + 1)
\]
3. **Combining factors:**
Now, combine the factors we found:
\[
3x^4 + 7x^3 + 4x^2 = x^2(3x + 4)(x + 1)
\]
So, the final factorization of \( 3x^4 + 7x^3 + 4x^2 \) is:
\[
x^2(3x + 4)(x + 1)
\]
The linear and quadratic factors are \( (3x + 4) \) (linear), \( (x + 1) \) (linear), and \( x^2 \) (which is a quadratic in the sense of having a square term).
Thus, one of the possible responses is:
**\( x^2(3x+4)(x+1) \)**.