For the polynomial below, -1 is a zero.

Express as a product of linear factors.

Ask by Hamilton Cummings. in the United States

Mar 07,2025

Question

For the polynomial below, -1 is a zero. $$ f(x)=x^{3}-5 x^{2}+x+7 $$ Express $$ f(x) $$ as a product of linear factors. $$ f(x)=\square $$

UpStudy AI · Accepted Answer

We are given  
$$
f(x)=x^{3}-5x^{2}+x+7
$$
and we know $$-1$$ is a zero. This tells us that $$(x + 1)$$ is a factor of $$f(x)$$.

### Step 1. Factor Out $$(x+1)$$

We can use synthetic division to divide $$f(x)$$ by $$(x+1)$$. The coefficients of $$f(x)$$ are $$1$$ (for $$x^3$$), $$-5$$ (for $$x^2$$), $$1$$ (for $$x$$), and $$7$$ (constant).

Set up the synthetic division with $$-1$$:

$$
\begin{array}{r|cccc}
-1 & 1 & -5 & 1 & 7 \
   &   & -1 & 6 & -7 \ \hline
   & 1 & -6 & 7 & 0 \
\end{array}
$$

- Bring down the $$1$$.  
- Multiply $$1$$ by $$-1$$ to get $$-1$$; add to $$-5$$ to obtain $$-6$$.  
- Multiply $$-6$$ by $$-1$$ to get $$6$$; add to $$1$$ to obtain $$7$$.  
- Multiply $$7$$ by $$-1$$ to get $$-7$$; add to $$7$$ to obtain $$0$$.

The quotient is  
$$
x^2 - 6x + 7.
$$

### Step 2. Factor the Quadratic $$x^2 - 6x + 7$$

To factor the quadratic, we use the quadratic formula:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
$$
where $$a=1$$, $$b=-6$$, and $$c=7$$. Plug in these values:
$$
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1} = \frac{6 \pm \sqrt{36 - 28}}{2} = \frac{6 \pm \sqrt{8}}{2}.
$$

Simplify $$\sqrt{8}$$ to $$2\sqrt{2}$$:
$$
x = \frac{6 \pm 2\sqrt{2}}{2} = 3 \pm \sqrt{2}.
$$

So, the zeros are $$x = 3+\sqrt{2}$$ and $$x = 3-\sqrt{2}$$. This tells us that the quadratic factors as:
$$
x^2 - 6x + 7 = \bigl(x - (3+\sqrt{2})\bigr)\bigl(x - (3-\sqrt{2})\bigr).
$$

### Step 3. Write $$f(x)$$ as a Product of Linear Factors

Combining the factor from Step 1 with the factored quadratic, we obtain:
$$
f(x) = (x+1)\bigl(x - (3+\sqrt{2})\bigr)\bigl(x - (3-\sqrt{2})\bigr).
$$

Thus, the final answer is:

$$
f(x) = (x+1)(x-(3+\sqrt{2}))(x-(3-\sqrt{2})).
$$
$$ f(x) = (x + 1)(x - (3 + \sqrt{2}))(x - (3 - \sqrt{2})) $$

For the polynomial below, -1 is a zero.

Express as a product of linear factors.

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Solución

Step 1. Factor Out

Step 2. Factor the Quadratic

Step 3. Write as a Product of Linear Factors

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