Question

Determine the linear and quadratic factors of the expression 2x4+19x3+35x2 .(1 point) Responses (2x+5)(x3+7x2) left parenthesis 2 x plus 5 right parenthesis left parenthesis x cubed plus 7 x squared right parenthesis (2x+5)(x+7) left parenthesis 2 x plus 5 right parenthesis left parenthesis x plus 7 right parenthesis (2x3+5x2)(x+7) left parenthesis 2 x cubed plus 5 x squared right parenthesis left parenthesis x plus 7 right parenthesis x2(2x+5)(x+7)

Answer 1

To find the linear and quadratic factors of the expression \(2x^4 + 19x^3 + 35x^2\), we can start by factoring out the common term.

1. Factor out \(x^2\) from the expression:
\[
2x^4 + 19x^3 + 35x^2 = x^2(2x^2 + 19x + 35)
\]

Next, we need to factor the quadratic \(2x^2 + 19x + 35\). We will look for two numbers that multiply to \(2 \cdot 35 = 70\) and add up to \(19\).

The numbers \(14\) and \(5\) meet these conditions since:
- \(14 \times 5 = 70\)
- \(14 + 5 = 19\)

Now we can use these values to split the middle term:
\[
2x^2 + 14x + 5x + 35 = (2x^2 + 14x) + (5x + 35)
\]

This can be factored by grouping:
\[
= 2x(x + 7) + 5(x + 7) = (2x + 5)(x + 7)
\]

So the complete factorization of the original expression \(2x^4 + 19x^3 + 35x^2\) is:
\[
x^2(2x + 5)(x + 7)
\]

Thus, the linear factors are \(2x + 5\) and \(x + 7\), while the quadratic portion is \(x^2\), which can also be represented as \((1)(x^2)\).

The correct response would therefore be:
\[
x^2(2x + 5)(x + 7) \text{ or simply } (2x + 5)(x + 7).
\]

However, from the choices you provided, the matching answer is:
\[
x^2(2x + 5)(x + 7).
\]

Here's the summary of factors:
- Linear factors: \( (2x + 5)(x + 7) \)
- Quadratic factor: \( x^2 \) or when put as a linear term it would reflect from \(x^2\).

Thus, confirm that you select the option reflecting these factors correctly.