For the polynomial below, -2 is a zero.

Express as a product of linear factors.

Ask by Newton Goodwin. in the United States

Mar 29,2025

Question

For the polynomial below, -2 is a zero. $$ h(x)=x^{3}-4 x^{2}-10 x+4 $$ Express $$ h(x) $$ as a product of linear factors. $$ h(x)=\square $$

UpStudy AI · Accepted Answer

First, we are given that $$ -2 $$ is a zero of

$$
h(x) = x^3 - 4x^2 - 10x + 4.
$$

This means that $$ x + 2 $$ is a factor of $$ h(x) $$.

### Step 1. Divide $$ h(x) $$ by $$ x + 2 $$

Use synthetic division with $$ -2 $$:

1. Write down the coefficients: $$1$$, $$-4$$, $$-10$$, $$4$$.
2. Bring down the first coefficient:
   $$
   \begin{array}{r|cccc}
   -2 & 1 & -4 & -10 & 4 \
   \hline
      & 1 &    &     &   \
   \end{array}
   $$
3. Multiply $$1$$ by $$-2$$ and add to $$-4$$:
   $$
   -2 \quad \text{(multiply: } 1 \times -2 = -2 \text{)}
   $$
   $$
   -4 + (-2) = -6
   $$
4. Multiply $$-6$$ by $$-2$$ and add to $$-10$$:
   $$
   12 \quad \text{(multiply: } -6 \times -2 = 12 \text{)}
   $$
   $$
   -10 + 12 = 2
   $$
5. Multiply $$2$$ by $$-2$$ and add to $$4$$:
   $$
   -4 \quad \text{(multiply: } 2 \times -2 = -4 \text{)}
   $$
   $$
   4 + (-4) = 0
   $$

The synthetic division table becomes:

$$
\begin{array}{r|cccc}
-2 & 1 & -4 & -10 & 4 \
   &   & -2 & 12  & -4 \
\hline
   & 1 & -6 & 2   & 0 \
\end{array}
$$

Thus, the quotient is

$$
x^2 - 6x + 2.
$$

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

Use the quadratic formula:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
$$

with $$ a = 1 $$, $$ b = -6 $$, and $$ c = 2 $$.

Compute the discriminant:

$$
b^2 - 4ac = (-6)^2 - 4(1)(2) = 36 - 8 = 28.
$$

Then,

$$
x = \frac{6 \pm \sqrt{28}}{2} = \frac{6 \pm 2\sqrt{7}}{2} = 3 \pm \sqrt{7}.
$$

So, the quadratic factors as

$$
\left(x - (3 + \sqrt{7})\right)\left(x - (3 - \sqrt{7})\right).
$$

### Step 3. Write $$ h(x) $$ as a product of linear factors

Including the factor from the synthetic division, we have:

$$
h(x) = (x + 2)\left(x - (3 + \sqrt{7})\right)\left(x - (3 - \sqrt{7})\right).
$$

Thus,

$$
h(x) = (x + 2)\left(x - (3 + \sqrt{7})\right)\left(x - (3 - \sqrt{7})\right).
$$
$$ h(x) = (x + 2)(x - (3 + \sqrt{7}))(x - (3 - \sqrt{7})) $$

For the polynomial below, -2 is a zero.

Express as a product of linear factors.

Upstudy AI Solution

Answer

Solution

Step 1. Divide by

Step 2. Factor the quadratic

Step 3. Write as a product of linear factors

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For the polynomial below, -2 is a zero. Express as a product of linear factors.