Question

fill in the blank 1 point
17
a function is shown below.
$h(x) = 4x^3 - 5x^2 - 23x + 6$
what is the distance, to the nearest hundredth of a unit, between the two zeros that are closest to each other?
distance=
1.5

Explanation:

Step1: Find rational zero via Rational Root Theorem

Possible rational zeros: $\pm1, \pm2, \pm3, \pm6, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}$. Test $x=2$:
$h(2)=4(2)^3 -5(2)^2 -23(2)+6=32-20-46+6=-28
eq0$
Test $x=3$:
$h(3)=4(27)-5(9)-23(3)+6=108-45-69+6=0$
So $x=3$ is a zero, factor $(x-3)$.

Step2: Polynomial division to factor

Divide $h(x)$ by $(x-3)$:
$$\frac{4x^3-5x^2-23x+6}{x-3}=4x^2+7x-2$$

Step3: Find roots of quadratic

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ for $4x^2+7x-2=0$, where $a=4, b=7, c=-2$:
$$x=\frac{-7\pm\sqrt{7^2-4(4)(-2)}}{2(4)}=\frac{-7\pm\sqrt{49+32}}{8}=\frac{-7\pm\sqrt{81}}{8}=\frac{-7\pm9}{8}$$
Calculate roots:
$x_1=\frac{-7+9}{8}=\frac{2}{8}=0.25$, $x_2=\frac{-7-9}{8}=\frac{-16}{8}=-2$

Step4: List all zeros and find closest

Zeros: $-2, 0.25, 3$
Calculate distances:
$|0.25 - (-2)|=2.25$, $|3-0.25|=2.75$, $|3-(-2)|=5$
The smallest distance is $2.25$.

Answer:

2.25