Quadratics & Polynomials - ACT Math

1. Factor $4x^2 - 9$. What is the expression?

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

2. For what discriminant values does $ax^2 + bx + c = 0$ have real and equal roots?

   Discriminant $b^2 - 4ac = 0$. Zero discriminant means one repeated real solution.

3. Factor $x^2 - 1$.

   $(x+1)(x-1)$. Difference of squares: $x^2 - 1^2 = (x+1)(x-1)$.

4. State the difference of squares factorization formula.

   $a^2-b^2=(a-b)(a+b)$. Special factoring pattern for expressions with squared terms.

5. State the factor theorem relating $f(a)$ to the factor $(x-a)$.

   $(x-a)$ is a factor iff $f(a)=0$. A polynomial has $(x-a)$ as a factor if and only if $f(a)=0$.

6. What are the roots of $x^2 - 9$?

   $x = 3, -3$. Difference of squares: $x^2 - 9 = (x-3)(x+3) = 0$.

7. What does $\Delta=0$ indicate about the real solutions of $ax^2+bx+c=0$?

   One real double solution. Zero discriminant means the parabola touches the x-axis at exactly one point.

8. What is the vertex form of a quadratic function?

   $y = a(x - h)^2 + k$. Form that shows vertex $(h,k)$ directly.

9. State the binomial theorem.

   $(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$. Formula for expanding $(x+y)^n$ using binomial coefficients.

10. Factor $x^2 - 9$ completely.

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

11. What is the result of multiplying $x(x+1)$?

    $x^2 + x$. Distribute: $x(x) + x(1) = x^2 + x$.

12. State the definition of the degree of a polynomial.

    Largest exponent of $x$ with nonzero coefficient. The highest power of the variable with a non-zero coefficient.

13. Simplify $2x^2 - 4x + 2$. What is the expression?

    $2(x^2 - 2x + 1)$. Factor out common factor of 2.

14. Which term is the quadratic term in $5x^2 - 3x + 8$?

    Quadratic term is $5x^2$. Term with $x^2$ (degree 2).

15. State Vieta's formulas for $ax^2+bx+c=0$ with roots $r_1,r_2$.

    $r_1+r_2=-\frac{b}{a}$ and $r_1r_2=\frac{c}{a}$. Relationships between coefficients and roots without solving the equation.

16. For what discriminant values does $ax^2 + bx + c = 0$ have complex roots?

    Discriminant $b^2 - 4ac < 0$. Negative discriminant means no real solutions.

17. Identify the constant term in $3x^2 - 5x + 7$.

    7. The term without a variable (degree 0).

18. What is the y-intercept of $y = 3x^2 + 2x - 4$?

    Y-intercept is $-4$. Y-intercept occurs when $x=0$, giving $y=c$.

19. Find $f(2)$ for $f(x) = x^2 - 3x + 2$.

    $f(2) = 0$. Substitute $x=2$: $(2)^2 - 3(2) + 2 = 4 - 6 + 2 = 0$.

20. Find the discriminant of $3x^2-4x+5=0$.

    $\Delta=-44$. Calculate $\Delta=(-4)^2-4(3)(5)=16-60=-44$.

21. For what discriminant values does $ax^2 + bx + c = 0$ have real and distinct roots?

    Discriminant $b^2 - 4ac > 0$. Positive discriminant means two different real solutions.

22. Factor $x^2 - 4x + 4$.

    $(x-2)^2$. Perfect square trinomial: $(x-2)^2 = x^2 - 4x + 4$.

23. Find the vertex of $y = x^2 - 4x + 4$.

    $(2, 0)$. Complete the square or use $x = -\frac{b}{2a} = 2$, $y = 0$.

24. State the standard form of a quadratic function in $x$.

    $f(x)=ax^2+bx+c$, with $a\ne 0$. General form where $a$ determines parabola direction and cannot be zero.

25. If $f(x)=x^2-1$, what are the zeros of $f(x)$?

    $x=-1$ and $x=1$. Set $f(x)=0$: $x^2-1=0$, so $(x-1)(x+1)=0$.

26. State the factor theorem.

    If $f(c) = 0$, then $(x - c)$ is a factor of $f(x)$. If $c$ is a root, then $(x-c)$ divides the polynomial.

27. Subtract $(3x^2 + 2x)$ from $(5x^2 + x + 4)$.

    $2x^2 - x + 4$. Subtract each term: $(5-3)x^2 + (1-2)x + (4-0)$.

28. Evaluate $x^2 - 4x + 4$ for $x = 2$.

    Result is $0$. Substitute $x=2$: $(2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0$.

29. What is the linear term in the polynomial $7x^2 + 2x - 1$?

    Linear term is $2x$. Term with $x$ (degree 1).

30. Identify the vertex form of a quadratic function.

    $y = a(x-h)^2 + k$. Form showing vertex $(h,k)$ and vertical stretch/compression $a$.

31. In $y=a(x-h)^2+k$, what are the vertex coordinates?

    Vertex $=(h,k)$. The vertex is at the point $(h,k)$ in this form.

32. Find the remainder when $x^3 - 2x^2 + x - 1$ is divided by $x - 1$.

    Remainder is $-1$. By remainder theorem: $f(1) = 1 - 2 + 1 - 1 = -1$.

33. Simplify $x^3 - x^2 + x - 1$ for $x = 1$.

    Result is $0$. Substitute $x=1$: $1 - 1 + 1 - 1 = 0$.

34. Determine the roots of $x^2 - 5x + 6 = 0$.

    Roots are $x = 2$ and $x = 3$. Factor as $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$.

35. How many real solutions does $x^2+4x+5=0$ have?

    Zero real solutions. Discriminant $\Delta=16-20=-4<0$, so no real solutions.

36. Simplify: $(x+2)(x-5)$.

    $x^2-3x-10$. Use FOIL: $(x+2)(x-5)=x^2-5x+2x-10$.

37. What is the product of $(x+2)$ and $(x-3)$?

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

38. State the factored form of a quadratic with roots $r_1$ and $r_2$.

    $a(x-r_1)(x-r_2)$. Form showing the quadratic as a product using its roots or zeros.

39. What is the constant term in the polynomial $x^3 + 4x^2 - 2x + 6$?

    Constant term is 6. Term with no variable (degree 0).

40. What is the discriminant of $ax^2+bx+c=0$?

    $\Delta=b^2-4ac$. Expression under the square root in the quadratic formula.

41. Solve for $x$: $x^2 - 5x + 6 = 0$

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

42. Determine if $x = -2$ is a root of $x^3 + 2x^2 - 5$.

    Not a root. Substitute $x=-2$: $(-2)^3 + 2(-2)^2 - 5 = -8 + 8 - 5 = -5 \neq 0$.

43. What is the sum of the coefficients in $2x^2 - 3x + 5$?

    4. Add all coefficients: $2 + (-3) + 5 = 4$.

44. Factor completely: $x^2+10x+25$.

    $(x+5)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.

45. State the zero-product property used when solving $(A)(B)=0$.

    $A=0$ or $B=0$. If a product equals zero, at least one factor must equal zero.

46. Find the vertex of $y=x^2-6x+5$.

    $(3,-4)$. Complete the square or use vertex formula $x=-\frac{b}{2a}=3$, then $y=5-9=-4$.

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

    5. The highest power of the variable in the polynomial.

48. Identify the cubic term in $4x^3 + 3x^2 + x + 5$.

    Cubic term is $4x^3$. Term with $x^3$ (degree 3).

49. Evaluate $f(2)$ for $f(x)=x^3-4x+1$.

    $1$. Substitute: $f(2)=2^3-4(2)+1=8-8+1=1$.

50. What is the vertex of $y = (x-2)^2 + 3$?

    Vertex is $(2, 3)$. In vertex form, $(h,k)$ gives the vertex coordinates.

51. State the quadratic formula.

    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Formula to solve any quadratic equation $ax^2 + bx + c = 0$.

52. Factor completely: $x^2-9$.

    $(x-3)(x+3)$. Difference of squares pattern: $a^2-b^2=(a-b)(a+b)$.

53. What does the sign of $a$ determine for $y=ax^2+bx+c$?

    Opens up if $a>0$; opens down if $a<0$. The coefficient $a$ determines whether the parabola opens upward or downward.

54. Expand and simplify: $(x-4)^2$.

    $x^2-8x+16$. Use the formula $(a-b)^2=a^2-2ab+b^2$.

55. Convert $y = 3(x-2)^2 + 1$ to standard form.

    $y = 3x^2 - 12x + 13$. Expand $ $(x-2)^2$ $ and multiply by 3, then add 1.

56. Simplify $(2x + 3)(x - 1)$.

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

57. What is the value of the polynomial $3x^2 - 2x + 1$ at $x = 1$?

    2. Substitute $x = 1$: $3(1)^2 - 2(1) + 1 = 2$.

58. State the maximum number of real zeros a degree $n$ polynomial can have.

    At most $n$ real zeros. A polynomial of degree $n$ can cross the x-axis at most $n$ times.

59. Identify the leading coefficient of $2x^3 - 4x^2 + 5x - 7$.

    Leading coefficient is 2. Coefficient of the highest degree term.

60. State the sum of cubes factorization formula.

    $a^3+b^3=(a+b)(a^2-ab+b^2)$. Special factoring pattern for the sum of two perfect cubes.

61. What is the end behavior of an even-degree polynomial with negative leading coefficient?

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

62. Simplify $(x + 1)(x - 1)$. What is the expression?

    $x^2 - 1$. Difference of squares formula.

63. State the axis of symmetry for $y=ax^2+bx+c$.

    $x=-\frac{b}{2a}$. The x-coordinate where the parabola reaches its maximum or minimum.

64. What is the expanded form of $(x + 3)^2$?

    $x^2 + 6x + 9$. Perfect square: $(a+b)^2 = a^2 + 2ab + b^2$.

65. What is the solution to $x^2 + 4x + 4 = 0$?

    $x = -2$. Perfect square: $(x+2)^2 = 0$, so $x = -2$.

66. What is the sum of the solutions to $2x^2 - 8x = 0$?

    Sum is $4$. Factor as $2x(x-4) = 0$, roots are $x=0,4$; sum is $0+4=4$.

67. What is the result of multiplying $(x + 2)(x - 3)$?

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

68. State the perfect square trinomial formula for $(a-b)^2$.

    $(a-b)^2=a^2-2ab+b^2$. Expansion formula for the square of a difference.

69. What does the discriminant indicate about the roots?

    Number and nature of roots. Positive: two real roots; zero: one root; negative: no real roots.

70. What is the coefficient of $x$ in $5x^2 + 3x - 7$?

    3. The numerical factor multiplying the $x$ term.

71. Find the discriminant of $ax^2 + bx + c = 0$.

    $b^2 - 4ac$. Expression under the square root in the quadratic formula.

72. Solve for $x$: $x^2 = 49$.

    $x = 7$ or $x = -7$. Take square root of both sides.

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

    Degree is 4. Highest power of the variable determines degree.

74. State the quadratic formula for solutions to $ax^2+bx+c=0$.

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square on the general quadratic equation.

75. Determine the x-intercepts of $y = x^2 - 4$.

    $x = 2$ and $x = -2$. Set $y=0$ and solve: $x^2 - 4 = 0$.

76. Find the greatest common factor of $2x^3$ and $4x^2$.

    $2x^2$. Factor out the highest power common to both terms.

77. Use Vieta: for $x^2-7x+10=0$, what is the product of the roots?

    $10$. Product of roots equals $\frac{c}{a}=\frac{10}{1}=10$.

78. State the vertex form of a quadratic function.

    $y=a(x-h)^2+k$. Form that directly shows the vertex coordinates $(h,k)$.

79. Find the sum of $(2x^2 + 3x + 4)$ and $(x^2 - 5x + 1)$.

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

80. Which term describes $a$ in $ax^2 + bx + c$?

    Coefficient of $x^2$. The leading coefficient determines parabola direction and width.

81. What is the sum of $(x^2 + x)$ and $(2x^2 - x + 3)$?

    $3x^2 + 3$. Combine like terms: $(1+2)x^2 + (1-1)x + (0+3)$.

82. What does $\Delta>0$ indicate about the real solutions of $ax^2+bx+c=0$?

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

83. Find the zero of the polynomial $x - 4$.

    $x = 4$. Set the polynomial equal to zero: $x - 4 = 0$.

84. What is the $y$-intercept of $y=ax^2+bx+c$?

    $(0,c)$. Set $x=0$ in the equation to find where the parabola crosses the y-axis.

85. If $p(x) = x^3 - 2x^2 + 4$, what is $p(1)$?

    $p(1) = 3$. Substitute $x=1$: $(1)^3 - 2(1)^2 + 4 = 1 - 2 + 4 = 3$.

86. What is the product of the roots of $ax^2 + bx + c = 0$?

    $\frac{c}{a}$. By Vieta's formulas for quadratic equations.

87. State the quadratic formula.

    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations when factoring fails.

88. State the difference of cubes factorization formula.

    $a^3-b^3=(a-b)(a^2+ab+b^2)$. Special factoring pattern for the difference of two perfect cubes.

89. What is the leading coefficient of a polynomial?

    Coefficient of the highest-degree term. The number multiplying the term with the highest degree.

90. What is the sum of the roots of $ax^2 + bx + c = 0$?

    $-\frac{b}{a}$. By Vieta's formulas for quadratic equations.

91. Identify the degree of $x^3 + 2x^2 - 5x + 1$.

    $3$. The highest power of $x$ in the polynomial is $3$.

92. Identify the axis of symmetry for $y = ax^2 + bx + c$.

    $x = \frac{-b}{2a}$. X-coordinate of the parabola's vertex and line of symmetry.

93. What does $\Delta<0$ indicate about the solutions of $ax^2+bx+c=0$?

    No real solutions (two complex solutions). Negative discriminant means the parabola doesn't cross the x-axis.

94. Which term describes the highest power of $x$ in a polynomial?

    Degree. The highest exponent determines polynomial classification.

95. What is the end behavior of an odd-degree polynomial with positive leading coefficient?

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

96. Convert $x^2 + 4x + 4$ to vertex form.

    $(x + 2)^2$. Perfect square trinomial factorization.

97. What is the standard form of a quadratic equation?

    $ax^2 + bx + c = 0$. General form where $a ≠ 0$ and equation equals zero.

98. Identify the leading coefficient in $4x^3 - x^2 + 5$.

    4. The coefficient of the term with the highest degree.

99. State the remainder theorem for dividing $f(x)$ by $(x-a)$.

    Remainder $=f(a)$. The remainder equals the function value at the divisor's root.

100. State the perfect square trinomial formula for $(a+b)^2$.

     $(a+b)^2=a^2+2ab+b^2$. Expansion formula for the square of a sum.

101. Solve for $x$: $x^2-5x+6=0$.

     $x=2$ or $x=3$. Factor as $(x-2)(x-3)=0$ using zero-product property.

102. What is the axis of symmetry for $y = ax^2 + bx + c$?

     $x = \frac{-b}{2a}$. Vertical line through the vertex of any parabola.

103. Identify the vertex $x$-coordinate of $y=2x^2-8x+1$.

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

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

     $(x + 2)(x + 3)$. Find factors of 6 that add to 5: 2 and 3.

105. Simplify $(2x - 3)^2$. What is the expression?

     $4x^2 - 12x + 9$. Perfect square trinomial: $(a-b)^2 = a^2 - 2ab + b^2$.

106. Solve for $x$: $2x^2+7x+3=0$.

     $x=-3$ or $x=-\frac{1}{2}$. Factor as $(2x+1)(x+3)=0$ and solve each factor.

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