Question 334·Hard·Math

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f(x)= (x - 4)(x + 1)^2 (x - k)^2

In the equation above, $k$ is a positive constant. When the graph of $y = f(x)$ is drawn in the $xy$ -plane, the graph has exactly 2 distinct $x$ -intercepts. Which of the following could be the value of $k$ ?

A factored polynomial shows where its graph can meet the $x$ -axis: set each factor equal to zero. Count different $x$ -values, not how many times a factor appears. If a parameter adds a root but the question requires fewer distinct intercepts, that root must match one already on the list. You can graph the resulting value in Desmos to check the count.

Hints

Hint 1
An $x$ -intercept occurs where $f(x)=0$ . By the zero-product rule, a product is zero when at least one factor is zero. What $x$ -value makes each factor in the given product zero?
Hint 2
A squared factor such as $(x+1)^2$ has the same zero as $x+1$ . Squaring doesn't create another location on the $x$ -axis. How many different roots do the factors give before any match?
Hint 3
The two fixed roots are different. A repeated root still gives one intercept even when the curve only touches the axis. When checking a candidate, how can you count the distinct places where the graph meets the axis?

Step-by-step

Count the roots from the factors
Step 1
Find the possible intercepts
An $x$ -intercept is where the graph meets the $x$ -axis, so $f(x)=0$ there. Use the zero-product rule: the product is zero when at least one factor is zero. Set each factor to zero:
$x-4=0\quad\text{or}\quad x+1=0\quad\text{or}\quad x-k=0.$
Solve each equation for $x$ :
$x=4\quad\text{or}\quad x=-1\quad\text{or}\quad x=k.$
Step 2
Make the third root repeat
The superscript $2$ repeats a factor, but it doesn't add another distinct intercept, meaning one at a different $x$ -value. The fixed roots $-1$ and $4$ already give two locations. To have exactly two, the root $k$ must match one of those fixed roots. Otherwise there would be three different intercepts.
Step 3
Count a touching point
A repeated root from $(x-k)^2$ can make the graph touch the $x$ -axis rather than cross it. Touching still counts as an intercept, so don't discard that root.
Step 4
Check the two intercepts
The problem says $k$ is positive, so its root cannot match $-1$ ; it must match $4$ . Set $k=4$ and type $k=4$ , then $y=(x-4)(x+1)^2(x-k)^2$ in Desmos. Click the points where the graph meets the $x$ -axis: $(-1,0)$ and $(4,0)$ . There are exactly two distinct $x$ -intercepts, so $4$ could be the value of $k$ . Choice C.

Question 334·Hard·Math

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