# How do you factor y=x^3+8x^2+17x+10 ?

May 17, 2016

To get started, you will have to find a factor. This can be long. Use the remainder theorem to test possible factors. The possible factors are given by the rational root theorem. See below for more details.

#### Explanation:

The rational root theorem states that in a polynomial ƒ(x) = qx^n + mx^(n - 1) + ... + px^0, the possible roots will be at $\frac{p}{q}$.

Therefore, we can state that the possible roots are at (factors of 10)/(factors of 1). Listing these, we get:

$\frac{\pm 1 , \pm 2 , \pm 5 , \pm 10}{\pm 1}$

Simplifying further:

$\pm 1 , \pm 2 , \pm 5 , \pm 10$

Here comes the part that can be long. The remainder theorem states that for any polynomial function $p \left(x\right)$ being divided by x - a), the remainder is given by evaluating $p \left(a\right)$. A factor, when evaluated, will give a final result, or a remainder, of 0, since by definition a factor is a number that evenly divides another.

When I do this, I tend to go from positive to negative, from smallest to largest. From experience, it is relatively rare for you not to find a factor by $\left(x + 2\right)$. Following my technique, we will start with $\left(x - 1\right)$ and then proceed with $\left(x + 1\right)$. Inspect the following proofs:

$y = {\left(1\right)}^{3} + 8 {\left(1\right)}^{2} + 17 \left(1\right) + 10$

$y = 1 + 8 + 17 + 10$

$y = 36$, therefore, $\left(x - 1\right)$ is not a factor of this polynomial.

$y = {\left(- 1\right)}^{3} + 8 {\left(- 1\right)}^{2} + 17 \left(- 1\right) + 10$

$y = - 1 + 8 - 17 + 10$

$y = 0$

Therefore, $\left(x + 1\right)$ is a factor of this polynomial.

We must now divide $\left({x}^{2} + 8 {x}^{2} + 17 x + 10\right)$ by $\left(x + 1\right)$ to see what is left over.

The following image shows the long division (you could have also used synthetic division).

As you can see, the quotient is ${x}^{2} + 7 x + 10$

This can easily be factored by finding two numbers in $y = a {x}^{2} + b x + c , a = 1$ that multiply to c and that add to b. Two numbers that do this are $5 \mathmr{and} 2$

Therefore,

$y = {x}^{3} + 8 {x}^{2} + 17 x + 10 = \left(x + 1\right) \left(x + 5\right) \left(x + 2\right)$

or

$y = \left(x + 1\right) \left(x + 5\right) \left(x + 2\right)$

Hopefully this helps!