Practice Set 5
10 Polynomial Challenge Questions
This set targets common EOC trouble spots with complex factoring, multiplicity, end behavior, multiple representations, missing factors, and equation writing from roots or graphs.
Practice Questions
EOC Challenge Set: Polynomial Functions
Push into upper-level EOC review with factor reasoning, graph interpretation, and multi-step equation building.
Question 1
For f(x) = x³ - 5x² + mx + 24, if (x + 2) is a factor, what is the value of m?
Desmos Move: Use the factor theorem by substituting x = -2. Desmos can confirm your answer by graphing the finished polynomial and checking that x = -2 is an x-intercept.
Question 2
Which equation could model the table of values shown?
| x | -2 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| p(x) | 0 | -24 | 0 | 16 | 0 |
Use the zeros first, then use a nonzero output to determine the scale factor.
Desmos Move: Use the zeros from the table first, then substitute x = 0 to determine the scale factor. Graph the choices in Desmos and compare them to the table values.
Question 3
A student says the graph shown can be modeled by f(x) = (x + 1)(x - 2). Which response best corrects the mistake?
The graph touches at one zero and crosses at the other, so the multiplicities are different.
Desmos Move: Graph y = (x + 1)(x - 2) and y = (x + 1)²(x - 2). Only one of them bounces at x = -1 instead of crossing.
Question 4
The least-degree polynomial with zeros -3, 1, 1, and 4 can be written as f(x) = a(x + 3)(x - 1)²(x - 4). If f(2) = -20, what is the value of a?
Desmos Move: Substitute x = 2 and use the given output to solve for a. Then graph the result to confirm the intercepts and the point.
Question 5
Which expression is the complete factorization over the real numbers of x⁴ - 13x² + 36?
Desmos Move: Treat the expression like a quadratic in x² first. Desmos can then help you confirm the real zeros at x = -3, -2, 2, and 3.
Question 6
A polynomial has odd degree, a negative leading coefficient, and 4 turning points. Which statement must be true?
Desmos Move: Turning points help you estimate the minimum degree, while the sign and parity of the leading term determine the end behavior.
Question 7
A polynomial can be written as p(x) = 2(x + 1)(x - 3)(x - k). If p(0) = 30, what is the value of k?
Desmos Move: Substitute x = 0 to use the y-intercept. That turns the problem into a quick equation for the missing zero.
Question 8
A graph crosses the x-axis at x = -4 and x = 2, touches the x-axis at x = 1, and has y-intercept -8. Which equation could model the graph?
Desmos Move: Build the factors from the zeros first, use multiplicity to place the square, and then test x = 0 to check the y-intercept.
Question 9
Which statement must be true based on the table of values shown?
| x | -3 | -1 | 0 | 1 |
|---|---|---|---|---|
| p(x) | 0 | 0 | -15 | 0 |
A table can identify zeros, but it does not automatically show multiplicity.
Desmos Move: A table can identify zeros, but it does not always show whether the graph crosses or only touches at those zeros. Graphing helps fill in that missing information.
Question 10
A student says, "Since the zeros are -2 and 1, the polynomial must be f(x) = (x + 2)(x - 1)." If the graph's y-intercept is 12, which response is best?
Desmos Move: Zeros determine the factors, but the y-intercept can reveal the missing scale factor. Test x = 0 in Desmos or by substitution.