For the polynomial below, -1 is a zero.

Express as a product of linear factors.

Ask by Haynes Bartlett. in the United States

Mar 29,2025

Question

For the polynomial below, -1 is a zero. $$ h(x)=x^{3}+7 x^{2}+40 x+34 $$ Express $$ h(x) $$ as a product of linear factors. $$ h(x)=\square $$

UpStudy AI · Accepted Answer

To express the polynomial $$ h(x) = x^{3} + 7x^{2} + 40x + 34 $$ as a product of linear factors, we need to factorize it using the given zero, -1.

Given that -1 is a zero of the polynomial, we can write:
$$ h(x) = (x + 1)(x^{2} + 6x + 34) $$

Now, we need to factorize the quadratic expression $$ x^{2} + 6x + 34 $$ further.

Let's find the factors of 34 that add up to 6:
$$ 34 = 1 \times 34 $$
$$ 34 = 2 \times 17 $$

Since 1 + 34 = 35 and 2 + 17 = 19, neither of these pairs adds up to 6. Therefore, the quadratic expression $$ x^{2} + 6x + 34 $$ cannot be factored into linear factors.

Therefore, the polynomial $$ h(x) $$ can be expressed as:
$$ h(x) = (x + 1)(x^{2} + 6x + 34) $$
$$ h(x) = (x + 1)(x^{2} + 6x + 34) $$

For the polynomial below, -1 is a zero.

Express as a product of linear factors.

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Extra Insights

-1 | 1 | 7 | 40 | 34
| | -1 | -6 | -34

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For the polynomial below, -1 is a zero. Express as a product of linear factors.