Question

select the correct answer.
find the factors of function ( f ).
( f(x) = x^4 - 5x^3 - 4x^2 + 20x )
based on the factors, which statement is true about the graph of function ( f )?
a. the graph crosses the ( x )-axis at the point ( (2, 0) ).
b. the graph crosses the ( x )-axis at the point ( (4, 0) ).
c. the graph crosses the ( x )-axis at the point ( (-5, 0) ).
d. the graph crosses the ( x )-axis at the point ( (-4, 0) ).

Explanation:

Step1: Factor by grouping

Group terms to factor:

$$\begin{align*} f(x) &= x^4 - 5x^3 - 4x^2 + 20x \\ &= x^3(x - 5) - 4x(x - 5) \end{align*}$$

Step2: Factor out common binomial

Extract $(x-5)$ from both terms:
$f(x) = (x^3 - 4x)(x - 5)$

Step3: Factor $x$ from first term

Take out $x$ from the cubic factor:
$f(x) = x(x^2 - 4)(x - 5)$

Step4: Factor difference of squares

Use $a^2-b^2=(a-b)(a+b)$ for $x^2-4$:
$f(x) = x(x - 2)(x + 2)(x - 5)$

Step5: Identify x-intercepts

Set $f(x)=0$ to find roots:
$x=0,\ x=2,\ x=-2,\ x=5$

Step6: Match with options

Check which option's point is a root.

Answer:

A. The graph crosses the x-axis at the point (2, 0)