Polynomial Functions - ACT Math

1. Identify the leading coefficient of $f(x)=-4x^6+2x-1$.

   Leading coefficient $=-4$. The coefficient of the highest degree term is the leading coefficient.

2. Identify the constant term in $3x^4 - 2x + 7$.

   The constant term is 7. The term without any variable.

3. What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is even and $a_n<0$?

   As $x\to\pm\infty$, $f(x)\to-\infty$. Even degree with negative leading coefficient creates downward parabola-like behavior.

4. What is the vertex $x$-coordinate of $f(x)=ax^2+bx+c$?

   $x_v=-\frac{b}{2a}$. The vertex $x$-coordinate is the axis of symmetry.

5. What happens at $x=r$ when a zero has odd multiplicity?

   The graph crosses the $x$-axis at $x=r$. Odd multiplicity means the graph passes through the $x$-axis.

6. Find the product of $2x$ and $(3x + 4)$.

   $6x^2 + 8x$. Distribute $2x$ to each term.

7. What is the degree of a constant polynomial?

   The degree is 0. A constant has no variables.

8. What does $\Delta=b^2-4ac>0$ imply about the roots of $ax^2+bx+c$?

   Two distinct real roots. Positive discriminant means the parabola crosses the $x$-axis twice.

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

   The leading coefficient is $-5$. The coefficient of the highest degree term.

10. How do you factor $x^2 - 9$?

    $(x + 3)(x - 3)$. Difference of squares: $x^2 - 3^2$.

11. Identify the leading coefficient of $7x^5 - 3x^3 + x$.

    The leading coefficient is 7. The coefficient of the highest degree term.

12. Find the result of multiplying $x^2$ by $3x$.

    $3x^3$. Multiply coefficients and add exponents.

13. Identify the multiplicity of the zero $x=2$ for $f(x)=(x-2)^3(x+1)$.

    Multiplicity $=3$. The exponent of $(x-2)$ gives the multiplicity.

14. Write the polynomial $2x^3 + 3x - x^3$ in standard form.

    $x^3 + 3x$. Combine like terms and order by degree.

15. What does $\Delta=b^2-4ac=0$ imply about the roots of $ax^2+bx+c$?

    One real double root. Zero discriminant means the parabola touches the $x$-axis once.

16. What is the degree of the polynomial $x^2y^3 + xy^2$?

    The degree is 5. Highest degree term has exponents $2+3=5$.

17. What is the leading coefficient of $f(x)=a_nx^n+\cdots+a_0$?

    Leading coefficient $=a_n$. The coefficient of the highest degree term determines the leading coefficient.

18. Identify the form: $ax^2 + bx + c$.

    Quadratic form. Standard form of a degree-2 polynomial.

19. What is the sum of the zeros of $ax^2+bx+c$ (counting multiplicity)?

    $r_1+r_2=-\frac{b}{a}$. Vieta's formula relates coefficients to sums of roots.

20. What is the formula for the sum of cubes?

    $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Factorization formula for $a^3 + b^3$.

21. What does it mean for $r$ to be a zero (root) of $f(x)$?

    $f(r)=0$. A zero is an $x$-value where the function equals zero.

22. What is the discriminant of $ax^2+bx+c$ used to classify real roots?

    $\Delta=b^2-4ac$. The discriminant determines the nature of quadratic roots.

23. What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is odd and $a_n>0$?

    As $x\to-\infty$, $f(x)\to-\infty$; $x\to\infty$, $f(x)\to\infty$. Odd degree with positive leading coefficient goes from bottom-left to top-right.

24. State the term for a polynomial with two terms.

    Binomial. A polynomial with exactly two terms.

25. State the formula for the difference of squares.

    $a^2 - b^2 = (a + b)(a - b)$. Factor pattern for $a^2 - b^2$.

26. How do you write $x^3 - 27$ as a product?

    $(x - 3)(x^2 + 3x + 9)$. Difference of cubes: $x^3 - 3^3$.

27. Subtract $4x + 7$ from $6x + 3$.

    $2x - 4$. $(6x + 3) - (4x + 7) = 2x - 4$.

28. Find the remainder when $f(x)=x^3-4x+1$ is divided by $(x-2)$.

    $f(2)=1$. By the Remainder Theorem, $f(2) = 8 - 8 + 1 = 1$.

29. Factor $x^2 - 4x + 4$ completely.

    $(x - 2)^2$. Perfect square trinomial pattern.

30. What is the maximum possible number of turning points of a degree $n$ polynomial?

    At most $n-1$ turning points. Each turning point reduces the maximum by one from the degree.

31. What is the degree of a polynomial written as $a_nx^n+\cdots+a_0$ with $a_n\ne 0$?

    Degree $=n$. The degree equals the highest power of $x$ when $a_n \neq 0$.

32. What is the maximum possible number of real zeros of a degree $n$ polynomial?

    At most $n$ real zeros. The Fundamental Theorem of Algebra limits real zeros to at most the degree.

33. What is the conjugate root rule for polynomials with real coefficients?

    If $a+bi$ is a root, then $a-bi$ is a root. Complex roots of real polynomials always come in conjugate pairs.

34. Find the axis of symmetry of $f(x)=2x^2-8x+1$.

    $x=2$. Use $x = -\frac{b}{2a} = -\frac{-8}{2(2)} = 2$.

35. Find the sum of $2x^3 + x^2$ and $3x^3 - 5x^2$.

    $5x^3 - 4x^2$. Combine like terms: $(2+3)x^3 + (1-5)x^2$.

36. Identify the type of polynomial: $x^3 - 2x^2 + x - 5$.

    Cubic polynomial. The highest degree is 3.

37. Identify whether $(x-3)$ is a factor of $f(x)=x^2-6x+9$.

    Yes, because $f(3)=0$. Since $f(3) = 9 - 18 + 9 = 0$, $(x-3)$ is a factor.

38. What is the product of the zeros of $ax^2+bx+c$ (counting multiplicity)?

    $r_1r_2=\frac{c}{a}$. Vieta's formula relates coefficients to products of roots.

39. What is the standard form of a polynomial?

    Terms are ordered by descending powers of $x$. Powers decrease from left to right.

40. What is the sum of $4x^2 + 3x + 1$ and $x^2 - x$?

    $5x^2 + 2x + 1$. Add like terms: $(4+1)x^2 + (3-1)x + 1$.

41. Find $f(0)$ for $f(x)=3x^4-5x+12$.

    $f(0)=12$. Substitute $x = 0$ to get the constant term.

42. What is the term for a polynomial with three terms?

    Trinomial. A polynomial with exactly three terms.

43. Identify the zeroes of $x^2 - 5x + 6$.

    $x = 2, x = 3$. Set $(x-2)(x-3) = 0$ and solve.

44. Factor $x^2 + 5x + 6$ completely.

    $(x + 2)(x + 3)$. Find factors of 6 that sum to 5.

45. What is the value of $f(0)$ for $f(x)=a_nx^n+\cdots+a_0$?

    $f(0)=a_0$. Substituting $x = 0$ eliminates all terms except the constant.

46. What is the sum of $x^2 - 3x + 2$ and $x^2 + x - 4$?

    $2x^2 - 2x - 2$. Add corresponding terms.

47. What is the axis of symmetry of $f(x)=ax^2+bx+c$?

    $x=-\frac{b}{2a}$. The axis of symmetry passes through the vertex of the parabola.

48. Identify whether the graph crosses or touches at $x=1$ for $f(x)=(x-1)^4(x+2)$.

    Touches and turns at $x=1$. Even multiplicity $(x-1)^4$ means the graph touches and turns.

49. What is the degree of the polynomial $5x^2y^3 + 3xy - 4$?

    The degree is 5. The sum of exponents in $x^2y^3$ is $2+3=5$.

50. What is the product of $x + 2$ and $x - 3$?

    $x^2 - x - 6$. Use FOIL: $x^2 + 2x - 3x - 6$.

51. What is the constant term in $x^3 + 2x + 5$?

    The constant term is 5. The term without any variable.

52. What is the quadratic formula for solutions of $ax^2+bx+c=0$?

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. The quadratic formula solves $ax^2 + bx + c = 0$ directly.

53. What is the multiplicity rule for a factor $(x-r)^k$ of $f(x)$?

    Zero $r$ has multiplicity $k$. The exponent of the factor equals the multiplicity of the zero.

54. What is the Rational Root Theorem for possible rational zeros of $f(x)$?

    $\frac{p}{q}$ with $p\mid a_0$ and $q\mid a_n$. Rational zeros must have numerator dividing constant term, denominator dividing leading coefficient.

55. What is the Factor Theorem stated in terms of $f(r)$ and $(x-r)$?

    $f(r)=0\iff (x-r)$ is a factor of $f(x)$. The Factor Theorem connects zeros to linear factors.

56. Identify the other complex root if $3+2i$ is a root of a real-coefficient polynomial.

    $3-2i$. Complex conjugate pairs occur in real-coefficient polynomials.

57. State the formula for the difference of cubes.

    $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Factorization formula for $a^3 - b^3$.

58. What is the standard form of a polynomial function in $x$ of degree $n$?

    $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, $a_n\ne 0$. Standard polynomial form with highest degree term first and nonzero leading coefficient.

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

    The leading term is $-2x^3$. The term with the highest degree.

60. What is the Remainder Theorem for dividing $f(x)$ by $(x-r)$?

    Remainder $=f(r)$. The Remainder Theorem evaluates the polynomial at the divisor's zero.

61. What is the term for the highest power in a polynomial?

    Degree. The highest power of the variable.

62. Find the end behavior of $f(x)=-2x^4+5x^2-1$ as $x\to\pm\infty$.

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

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

    The degree is 4. The highest power of $x$ is 4.

64. Find the discriminant of $f(x)=x^2-6x+13$ and classify the real roots.

    $\Delta=-16$, no real roots. Calculate $\Delta = 36 - 52 = -16 < 0$, so no real roots.

65. Identify the degree of $f(x)=7x^5-3x^2+9$.

    Degree $=5$. The highest power term determines the degree.

66. Express $a^2 - 2ab + b^2$ as a product.

    $(a - b)^2$. Perfect square trinomial pattern.

67. What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is even and $a_n>0$?

    As $x\to\pm\infty$, $f(x)\to\infty$. Even degree with positive leading coefficient creates upward parabola-like behavior.

68. Find the roots of the polynomial $x^2 - x - 6$.

    $x = 3, x = -2$. Factor as $(x-3)(x+2) = 0$.

69. What is the greatest common factor of $4x^3$ and $6x^2$?

    $2x^2$. Factor out $2x^2$ from both terms.

70. Identify the possible rational zeros of $f(x)=2x^3-3x^2-8x+12$.

    $\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12,\pm\frac{1}{2},\pm\frac{3}{2}$. Use $\pm\frac{p}{q}$ where $p$ divides $12$ and $q$ divides $2$.

71. What is the product of $x + 1$ and $x + 2$?

    $x^2 + 3x + 2$. Use FOIL: $x^2 + 2x + x + 2$.

72. What is the $y$-intercept of $f(x)$ in terms of $f(0)$?

    $y$-intercept $=(0,f(0))$. The $y$-intercept occurs where $x = 0$.

73. Find the maximum number of turning points for a degree $6$ polynomial.

    At most $5$ turning points. A degree $6$ polynomial has at most $6-1=5$ turning points.

74. What is the end behavior of $f(x)=a_nx^n+\cdots$ when $n$ is odd and $a_n<0$?

    As $x\to-\infty$, $f(x)\to\infty$; $x\to\infty$, $f(x)\to-\infty$. Odd degree with negative leading coefficient goes from top-left to bottom-right.

75. What is the constant term of $f(x)=a_nx^n+\cdots+a_0$?

    Constant term $=a_0$. The term without $x$ is the constant term.

76. What does $\Delta=b^2-4ac<0$ imply about the roots of $ax^2+bx+c$?

    No real roots (two complex roots). Negative discriminant means the parabola doesn't touch the $x$-axis.

77. What happens at $x=r$ when a zero has even multiplicity?

    The graph touches and turns at $x=r$. Even multiplicity means the graph bounces off the $x$-axis.

78. What is the term for a polynomial with one term?

    Monomial. A polynomial with exactly one term.

79. What is the sum of the roots of $x^2 - 4x + 3$?

    The sum is 4. For $x^2 - 4x + 3$, sum equals $-b/a = 4$.

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