The polynomial function is defined by Which of the following points in the -plane is NOT an -intercept of the graph of ?

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SAT Nonlinear Functions Question 86 | Free SAT Prep | Aniko

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Question 86·Medium·Nonlinear Functions

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The polynomial function $p$ is defined by

p(x)=x^4 - 5x^3 + 6x^2.

Which of the following points in the $xy$ -plane is NOT an $x$ -intercept of the graph of $p$ ?

For questions asking which point is (or is not) an $x$ -intercept, remember that $x$ -intercepts occur where $y=0$ , so quickly either (1) set the function equal to $0$ and solve—often by factoring when you see common factors or a quadratic—or (2) plug in the $x$ -values from the answer choices to see which ones make the function value $0$ . Choose the faster method: factoring is efficient when the polynomial is nicely factorable, while direct substitution is better when there are only a few choices and the arithmetic is simple.

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Connect x-intercepts to the function value

For a point $(a,0)$ to be on the graph of $y=p(x)$ , what must be true about $p(a)$ ?

Use the equation $p(x)=0$

Write the equation $p(x)=0$ using $p(x)=x^4 - 5x^3 + 6x^2$ , and think about how you might factor it to find the $x$ -values that make this true.

Compare the roots with the choices

Once you know the $x$ -values that solve $p(x)=0$ , match them to the $x$ -coordinates in the answer choices and see which one does not appear.

Desmos Guide

Enter the function

In Desmos, type the function as y = x^4 - 5x^3 + 6x^2 so you can see its graph.

Check the candidate x-values

Create a table (click the + and choose "Table"), and in the $x$ -column enter the values $0$ , $2$ , $3$ , and $5$ . Desmos will automatically fill in the corresponding $y$ -values using your function.

Identify which point is not on the x-axis

Look at the $y$ -values in the table: the $x$ -values that produce $y=0$ correspond to $x$ -intercepts, and the one that produces a $y$ -value different from $0$ corresponds to the point that is not an $x$ -intercept in the answer choices.

Step-by-step Explanation

Understand what an x-intercept means

An $x$ -intercept of the graph of $p$ is a point where the graph crosses or touches the $x$ -axis. On the $x$ -axis, the $y$ -coordinate is $0$ , so for an $x$ -intercept $(a,0)$ we must have $p(a)=0$ . In this problem, each answer choice is a point $(x,0)$ , so we are looking for which $x$ -value does not make $p(x)$ equal to $0$ .

Solve $p(x)=0$ by factoring the polynomial

We want to find all $x$ such that $p(x)=0$ .

Start with the given function:

p(x)=x^4 - 5x^3 + 6x^2.

Factor out the greatest common factor $x^2$ :

p(x)=x^2(x^2 - 5x + 6).

Now factor the quadratic $x^2 - 5x + 6$ :

x^2 - 5x + 6 = (x-2)(x-3).

So the fully factored form is

p(x)=x^2(x-2)(x-3).

The solutions to $p(x)=0$ are the $x$ -values that make at least one factor equal to $0$ .

Match the solutions with the answer choices

From the factored form $p(x)=x^2(x-2)(x-3)$ , the equation $p(x)=0$ is satisfied when $x^2=0$ , $x-2=0$ , or $x-3=0$ , which gives $x=0$ , $x=2$ , and $x=3$ .

These correspond to the points $(0,0)$ , $(2,0)$ , and $(3,0)$ , which are $x$ -intercepts of the graph. The only answer choice that does not correspond to a solution of $p(x)=0$ is $(5,0)$ , so $(5,0)$ is not an $x$ -intercept.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation