Practice Set 3
10 Strategic Questions
This set focuses on modeling, error analysis, comparing representations, multiplicity, efficient solution strategies, and multi-step polynomial reasoning.
Practice Questions
DOK 3 Practice Set: Polynomial Reasoning
Use Desmos strategically to test claims, compare representations, and justify multi-step decisions in richer polynomial questions.
Question 1
A student says the graph shown could be modeled by f(x) = (x + 2)²(x - 1) because the graph touches at x = -2 and crosses at x = 1. Which response best explains the mistake?
Use the intercept behavior and the end behavior together before choosing an equation.
Desmos Move: Graph y = (x + 2)²(x - 1) and y = -(x + 2)²(x - 1). The intercept behavior is the same, but the end behavior changes.
Question 2
Which least-degree polynomial fits all of these conditions: zero at x = -3, double zero at x = 2, and point (0, 24)?
Desmos Move: Start with the factor structure from the zeros, then use the point to solve for the scale factor a in a(x + 3)(x - 2)².
Question 3
A table of values is shown for a polynomial p(x). A student claims p(x) = 2(x + 2)(x - 2) because the zeros are x = -2 and x = 2. Which response is best?
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| p(x) | 0 | 6 | 8 | 6 | 0 |
The zeros matter, but the sign of the outputs between the zeros matters too.
Desmos Move: Compare y = 2(x + 2)(x - 2) and y = -2(x + 2)(x - 2). Check which one matches the positive values between x = -2 and x = 2.
Question 4
A school store models monthly profit with P(x) = -2x² + 48x - 160, where x is the number of spirit shirts sold. Which recommendation best uses the model?
Desmos Move: Graph the quadratic, then use the x-intercepts for break-even points and the vertex for the maximum profit.
Question 5
Jalen says, "Since f(2) = 0 for f(x) = x³ - 5x² + 2x + 8, the factor must be (x + 2)." Which response is most accurate?
Desmos Move: Use the remainder theorem carefully: if f(c) = 0, then (x - c) is a factor. Desmos can help confirm the zeros at x = -1, 2, and 4.
Question 6
Two students analyze h(x) = x⁴ - 5x² + 4. Student A says the function has 2 real zeros because it looks like a quadratic in x². Student B says the function has 4 real zeros because it factors into (x² - 1)(x² - 4). Which evaluation is correct?
Desmos Move: Rewrite the quartic as a quadratic in x², then graph it to verify all four x-intercepts.
Question 7
A polynomial graph has 3 turning points and both ends fall. Which statement must be true?
Desmos Move: Use turning points to think about minimum possible degree, then use end behavior to determine whether the degree is even or odd and whether the leading coefficient is positive or negative.
Question 8
The polynomial p(x) = x³ - kx² - 4x + 4k has zeros at x = 2 and x = -1. What is the value of k?
Desmos Move: Since x = 2 is always a zero, substitute x = -1 into the expression and solve for k. Then graph the result to confirm the intercepts.
Question 9
A student wants all real zeros of f(x) = x⁴ - 6x² + 5. Which strategy is most efficient, and why?
Desmos Move: This is a great calculator-active question: use algebraic substitution first, then confirm the four x-intercepts on the graph.
Question 10
A graph crosses the x-axis at x = -4 and x = 3 and only touches the x-axis at x = 1. It also passes through (0, -12). Which equation could model the graph?
Desmos Move: Use the intercept behavior first to build the factors. Then test x = 0 to check which equation matches the y-intercept.