Question

what are the linear and quadratic factors of the expression 3x^4+7x^3+4x^2

Answer 1

To factor the expression \( 3x^4 + 7x^3 + 4x^2 \), we can start by factoring out the greatest common factor (GCF) of the terms.

The GCF of the coefficients \( 3, 7, \) and \( 4 \) is \( 1 \), and the smallest power of \( x \) present in all terms is \( x^2 \). Thus, we can factor out \( x^2 \):

\[
3x^4 + 7x^3 + 4x^2 = x^2(3x^2 + 7x + 4)
\]

Next, we need to factor the quadratic expression \( 3x^2 + 7x + 4 \). To do this, we can look for two numbers that multiply to \( 3 \times 4 = 12 \) and add to \( 7 \). The numbers \( 3 \) and \( 4 \) meet this requirement.

Using them, we can rewrite the quadratic:

\[
3x^2 + 3x + 4x + 4
\]

Now, we can group the terms:

\[
(3x^2 + 3x) + (4x + 4)
\]

Factoring by grouping, we get:

\[
3x(x + 1) + 4(x + 1) = (3x + 4)(x + 1)
\]

Putting it all together, we find:

\[
3x^4 + 7x^3 + 4x^2 = x^2(3x + 4)(x + 1)
\]

Thus, the linear and quadratic factors of the expression \( 3x^4 + 7x^3 + 4x^2 \) are:

- Linear factors: \( (3x + 4) \) and \( (x + 1) \)
- Quadratic factor: \( x^2 \) (since it's a degree 2 factor, not linear but still an important factor of the original expression).

Final factorization:

\[
x^2(3x + 4)(x + 1)
\]