Question

find the factors of function f.
f(x) = x⁴ − 5x³ − 4x² + 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 (4, 0).
○ b. the graph crosses the x-axis at the point (2, 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 the function

First, factor out the common factor \( x \) from \( f(x) = x^4 - 5x^3 - 4x^2 + 20x \):
\( f(x)=x(x^3 - 5x^2 - 4x + 20) \)
Then, group the terms in the cubic polynomial:
\( x^3 - 5x^2 - 4x + 20=x^2(x - 5)-4(x - 5) \)
Factor out \( (x - 5) \):
\( x^2(x - 5)-4(x - 5)=(x^2 - 4)(x - 5) \)
And factor \( x^2 - 4 \) as a difference of squares:
\( x^2 - 4=(x - 2)(x + 2) \)
So the fully factored form of \( f(x) \) is:
\( f(x)=x(x - 2)(x + 2)(x - 5) \)

Step2: Find the x - intercepts

The x - intercepts of a function \( y = f(x) \) are found by setting \( f(x)=0 \).
Setting \( f(x)=0 \), we have \( x(x - 2)(x + 2)(x - 5)=0 \)
Using the zero - product property (if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \)), we get the solutions \( x = 0 \), \( x=2 \), \( x=-2 \), and \( x = 5 \)

Step3: Analyze the options

• Option A: The x - intercepts are \( x = 0,2,-2,5 \). There is no x - intercept at \( x = 4 \), so the graph does not cross the x - axis at \( (4,0) \).
• Option B: When \( x = 2 \), \( f(2)=0 \), so the graph crosses the x - axis at \( (2,0) \).
• Option C: The x - intercepts are \( x = 0,2,-2,5 \). There is no x - intercept at \( x=-5 \), so the graph does not cross the x - axis at \( (-5,0) \).
• Option D: The x - intercepts are \( x = 0,2,-2,5 \). There is no x - intercept at \( x=-4 \), so the graph does not cross the x - axis at \( (-4,0) \).

Answer:

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