Polynomial Functions and Rates of Change - AP Precalculus

1. State the derivative of $5x^3 - 2x^2 + x - 8$.

   $15x^2 - 4x + 1$. Apply power rule to each term separately.

2. Find the roots of $x^2 - 4x + 4 = 0$.

   $x = 2$ (multiplicity 2). Perfect square: $(x-2)^2 = 0$ gives repeated root.

3. What is the end behavior of $f(x) = -3x^2 + 5x$?

   As $x \to \pm \infty$, $f(x) \to -\infty$. Even degree with negative leading coefficient.

4. Determine the polynomial degree of $7x^5 - 4x^3 + x - 9$.

   5. The highest power of $x$ is 5.

5. What is the leading coefficient of $x^4 - 3x^3 + 5x - 2$?

   1. Coefficient of highest degree term when no coefficient shown is 1.

6. What is the degree of the polynomial $3x^4 + 5x^2 - 7$?

   4. The highest power of $x$ determines the degree.

7. Which term is the cubic term in $7x^3 + 2x^2 - x + 5$?

   $7x^3$. The term containing $x$ to the third power.

8. Identify the leading coefficient of $6x^3 - 4x + 2$.

   6. The coefficient of the highest degree term.

9. Find the derivative of $f(x) = x^3 + 2x^2 - x + 1$.

   $3x^2 + 4x - 1$. Apply power rule to each term.

10. What is the rate of change of $f(x) = x^2 + 3$ at $x = 2$?

    4. Evaluate $f'(2) = 2(2) = 4$ where $f'(x) = 2x$.

11. Identify the quadratic term in $x^2 - 4x + 4$.

    $x^2$. The term containing $x$ to the second power.

12. Which polynomial has a degree of 0?

    Constant polynomials. Non-zero constants have degree 0.

13. What does it mean for a polynomial to be monic?

    Leading coefficient is 1. A monic polynomial has leading coefficient equal to 1.

14. What is the derivative of $f(x) = 7x^4 - 2$?

    $28x^3$. Apply power rule: $4 \cdot 7x^3 = 28x^3$.

15. What is the sum of the roots of $x^2 - 5x + 6 = 0$?

    5. For $ax^2 + bx + c = 0$, sum of roots is $-\frac{b}{a}$.

16. Which term in $5x^3 + 2x^2 - x + 7$ is the constant term?

    7. The term with no variable (degree 0).

17. What is the standard form of a polynomial?

    $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$. Terms arranged by decreasing powers of $x$.

18. What is the leading term of $-5x^4 + 3x^2 - x + 8$?

    $-5x^4$. The term with highest power and its coefficient.

19. What is the root multiplicity of $f(x) = (x-2)^3$ at $x=2$?

    3. The exponent indicates how many times the root repeats.

20. What is the constant term of $4x^3 + 9x - 7$?

    -7. The term with no variable attached.

21. Determine the degree of $f(x) = (x+1)(x-1)(x+3)$.

    3. Product of three linear factors gives degree 3.

22. What is the coefficient of $x^2$ in $3x^2 + 4x + 5$?

    3. The numerical factor multiplying $x^2$.

23. What is the vertex form of the quadratic $x^2 + 6x + 9$?

    $(x + 3)^2$. Complete the square or recognize perfect square trinomial.

24. Find the derivative of $f(x) = 3x^2 - 4x + 6$ at $x = 1$.

    2. Evaluate $f'(1) = 6(1) - 4 = 2$ where $f'(x) = 6x - 4$.

25. Find the sum of coefficients in $f(x) = 2x^3 - 4x + 6$.

    4. Evaluate $f(1) = 2(1)^3 - 4(1) + 6 = 4$.

26. Determine the y-intercept of $f(x) = x^3 - 4x + 7$.

    7. Evaluate $f(0) = 0^3 - 4(0) + 7 = 7$.

27. Find the $x$-intercepts of $f(x) = x^2 - 4$.

    $x = 2, x = -2$. Set $f(x) = 0$ and solve for $x$.

28. State the formula for the derivative of $ax^n$.

    $nax^{n-1}$. Power rule: bring down exponent, reduce power by 1.

29. What is the end behavior of $f(x) = 2x^3 - 5x + 1$?

    As $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$. Odd degree with positive leading coefficient.

30. Find the product of the roots of $x^2 - 5x + 6 = 0$.

    6. For $ax^2 + bx + c = 0$, product of roots is $\frac{c}{a}$.

31. Convert $f(x) = (x-1)(x+2)$ to standard form.

    $x^2 + x - 2$. Use FOIL to expand the product.

32. What is the constant term of $(x+2)(x-3)$?

    -6. Expand and identify the constant term.

33. Determine the y-intercept of $f(x) = 7x^2 - 3x + 5$.

    5. Set $x = 0$ and evaluate the function.

34. Determine the leading term of $2x^4 + 3x^3 - x + 5$.

    $2x^4$. The term with the highest power of $x$.

35. Identify the polynomial with zero leading coefficient.

    Not possible; $a_n \neq 0$ for leading term. A polynomial must have a non-zero leading coefficient.

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