# If b is the largest zero of x^3-5x^2-x+5 then which of the following quadratic equations is it also a zero of?

## 1) ${x}^{2} - 3 x - 10 = 0$ 2) ${x}^{2} + 3 x - 10 = 0$ 3) ${x}^{2} - 3 x + 10 = 0$ 4) ${x}^{2} + 3 x + 10 = 0$

May 4, 2016

1) ${x}^{2} - 3 x - 10 = 0$

#### Explanation:

First factor ${x}^{3} - 5 {x}^{2} - x + 5$ by grouping to find its zeros:

${x}^{3} - 5 {x}^{2} - x + 5$

$= \left({x}^{3} - 5 {x}^{2}\right) - \left(x - 5\right)$

$= {x}^{2} \left(x - 5\right) - 1 \left(x - 5\right)$

$= \left({x}^{2} - 1\right) \left(x - 5\right)$

$= \left(x - 1\right) \left(x + 1\right) \left(x - 5\right)$

which has zeros: $1$, $- 1$, $5$

So $b = 5$

Substituting this value of $b$ for $x$ in 1) we find:

${x}^{2} - 3 x - 10 = {5}^{2} - \left(3 \cdot 5\right) - 10 = 25 - 15 - 10 = 0$

So the answer is 1) ${x}^{2} - 3 x - 10 = 0$