Polynomial Equations - PSAT Math

1. What does the discriminant value $b^2-4ac>0$ imply about real solutions?

   Two distinct real solutions. Positive discriminant yields two different real roots.

2. What is the sum of the roots of $x^2-7x+10=0$?

   $7$. For $ax^2+bx+c=0$, sum of roots equals $-\frac{b}{a}$.

3. Evaluate $P(2)$ for $P(x)=x^3-4x$: what is $P(2)$?

   $0$. $P(2)=2^3-4(2)=8-8=0$.

4. What is the factored form of $x^2-9$?

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

5. Factor completely: $x^2+6x+9$.

   $(x+3)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.

6. Identify the solutions of $x^2-9=0$.

   $x=\pm 3$. Factor as $(x-3)(x+3)=0$, giving $x=3$ or $x=-3$.

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

   $b^2-4ac$. Determines the nature and number of solutions.

8. What is the factorization of $x^2-10x+25$?

   $(x-5)^2$. Recognize pattern: $(-5)^2 = 25$ and $2(-5) = -10$.

9. Solve $x^3-4x^2+4x=0$.

   $x=0$ or $x=2$. Factor out $x$: $x(x^2-4x+4)=x(x-2)^2=0$.

10. What is the zero-product property used to solve $(x-3)(x+5)=0$?

    If $ab=0$, then $a=0$ or $b=0$. A product equals zero only if at least one factor is zero.

11. What is the discriminant of $ax^2+bx+c=0$, and what does it determine?

    $b^2-4ac$; it determines the number of real solutions. Positive discriminant means 2 real solutions, zero means 1, negative means 0.

12. What is the discriminant for $ax^2+bx+c=0$, and what does it determine?

    Discriminant $b^2-4ac$; it determines the number of real solutions. Positive means 2 real roots, zero means 1, negative means 0.

13. What are the solutions of $(x-3)(x+5)=0$?

    $x=3$ or $x=-5$. Apply zero-product property: set each factor to zero.

14. Identify the solutions of $(x-3)(x+5)=0$.

    $x=3$ or $x=-5$. Set each factor to zero: $x-3=0$ gives $x=3$, $x+5=0$ gives $x=-5$.

15. What is the factorization pattern for a perfect square trinomial $a^2+2ab+b^2$?

    $a^2+2ab+b^2=(a+b)^2$. Perfect square trinomial factors to squared binomial.

16. Identify the number of real solutions to $x^2-6x+9=0$.

    $1$ real solution. $(x-3)^2=0$ has repeated root.

17. What is the maximum number of real solutions a degree $n$ polynomial equation can have?

    At most $n$ real solutions. By the Fundamental Theorem of Algebra, counting multiplicities.

18. Factor completely: $x^2-5x-6$.

    $(x-6)(x+1)$. Find two numbers that multiply to $-6$ and add to $-5$.

19. What is the Factor Theorem stated using $f(r)$ and $(x-r)$?

    $f(r)=0$ if and only if $(x-r)$ is a factor of $f(x)$. Root $r$ means $(x-r)$ divides $f(x)$.

20. What is the factorization pattern for a perfect square trinomial $a^2-2ab+b^2$?

    $a^2-2ab+b^2=(a-b)^2$. Perfect square with subtraction factors to squared difference.

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

    $0$ real solutions. Discriminant $=16-20=-4<0$, so no real solutions exist.

22. What is the standard form of a polynomial equation in $x$ of degree $n$?

    $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, $a_n\neq 0$. Standard form has terms in descending degree order with leading coefficient non-zero.

23. What is the degree of the polynomial $7x^4-3x^2+x-9$?

    Degree $4$. The degree is the highest exponent of the variable.

24. Solve the equation $(x-3)(x+5)=0$.

    $x=3$ or $x=-5$. Apply zero-product property: set each factor equal to zero.

25. Identify the remainder when $p(x)=x^3-4x+1$ is divided by $(x-2)$.

    $1$. By Remainder Theorem: $p(2)=2^3-4(2)+1=8-8+1=1$.

26. Identify the solutions of $x^2=49$.

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

27. How many real solutions does $x^2+4x+10=0$ have?

    $0$ real solutions. Discriminant $=16-40=-24<0$, so no real solutions.

28. What is the leading coefficient of $-5x^3+2x-1$?

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

29. What is the relationship between a factor and a root (Factor Theorem)?

    $(x-r)$ is a factor iff $f(r)=0$. A polynomial has factor $(x-r)$ exactly when $r$ is a root.

30. What is the factored form of the difference of squares $a^2-b^2$?

    $(a-b)(a+b)$. This is the difference of squares factorization pattern.

31. If $x=3$ is a solution of $P(x)=0$, what factor must $P(x)$ have?

    Factor $(x-3)$. By the Factor Theorem, if $P(3)=0$, then $(x-3)$ divides $P(x)$.

32. What does the Remainder Theorem say is the remainder when dividing $f(x)$ by $(x-c)$?

    Remainder $=f(c)$. When dividing $f(x)$ by $(x-c)$, the remainder equals $f(c)$.

33. Solve for $x$: $x^3-4x^2=0$.

    $x=0$ or $x=4$. Factor out $x^2$: $x^2(x-4)=0$, so $x=0$ (double root) or $x=4$.

34. What is the factored form of the perfect square trinomial $a^2-2ab+b^2$?

    $(a-b)^2$. Perfect square with negative middle term factors as $(a-b)$ squared.

35. What is the zero-product property used to solve factored equations?

    If $ab=0$, then $a=0$ or $b=0$. At least one factor must equal zero for the product to be zero.

36. Identify the solutions of $(x-4)(x+1)=0$.

    $x=4$ and $x=-1$. Set each factor to zero: $x-4=0$ gives $x=4$, $x+1=0$ gives $x=-1$.

37. What is the standard form of a polynomial in descending powers of $x$?

    $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. Terms arranged from highest to lowest degree.

38. Solve $x^2-5x+6=0$.

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

39. Identify the solutions of $x^2+2x-3=0$.

    $x=1$ and $x=-3$. Factor as $(x-1)(x+3)=0$.

40. What is the product of the roots of $3x^2+6x-9=0$?

    $\frac{c}{a}=\frac{-9}{3}=-3$. For $ax^2+bx+c=0$, product of roots equals $\frac{c}{a}$.

41. What is the degree of the polynomial $7x^5-3x^2+9$?

    $5$. The degree is the highest power of $x$ in the polynomial.

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

    $x=2$ or $x=3$. Factors to $(x-2)(x-3)=0$, giving these two solutions.

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

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

44. What is the definition of a polynomial in $x$?

    An expression $a_nx^n+\cdots+a_1x+a_0$ with $n$ a nonnegative integer. Each term has form $a_ix^i$ where $i$ ranges from 0 to $n$.

45. Use the Factor Theorem: if $P(2)=0$, what factor divides $P(x)$?

    $(x-2)$. By Factor Theorem, $P(2)=0$ means $(x-2)$ is a factor.

46. Identify the zeros of $f(x)=(x-2)(x+5)$.

    $x=2$ and $x=-5$. Set each factor equal to zero: $(x-2)=0$ and $(x+5)=0$.

47. What does it mean for $r$ to be a solution (root) of $p(x)=0$?

    $p(r)=0$. When $r$ is substituted for $x$, the polynomial equals zero.

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

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

49. Find the sum of the solutions of $x^2-7x+10=0$ without solving fully.

    $7$. By Vieta's formulas, sum of roots equals negative coefficient of $x$ divided by leading coefficient.

50. How many real solutions does $x^2+4x+8=0$ have?

    $0$ real solutions. Discriminant $=16-32=-16<0$, so no real solutions.

51. Solve $x^2-5x+6=0$.

    $x=2$ or $x=3$. Factor as $(x-2)(x-3)=0$, then solve each factor.

52. What is the factored form of $x^2+6x+9$?

    $(x+3)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.

53. Solve for $x$: $x^3-4x=0$.

    $x=-2$, $x=0$, or $x=2$. Factor as $x(x^2-4)=x(x-2)(x+2)=0$.

54. State the zero-product property used to solve factored equations.

    If $ab=0$, then $a=0$ or $b=0$. A product equals zero only if at least one factor is zero.

55. What is the factored form of $x^2+6x+9$?

    $(x+3)^2$. Perfect square trinomial: $a^2+2ab+b^2=(a+b)^2$.

56. What is the factored form of the perfect square trinomial $a^2+2ab+b^2$?

    $(a+b)^2$. Perfect square trinomial: first term squared plus twice the product plus last squared.

57. What is $P(2)$ for $P(x)=x^3-4x$?

    $0$. Substitute: $P(2) = 2^3 - 4(2) = 8 - 8 = 0$.

58. What does a discriminant $b^2-4ac<0$ imply about real solutions?

    No real solutions. Negative discriminant means no real square roots exist.

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

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Solves any quadratic equation when factoring is difficult.

60. What is $f(2)$ for $f(x)=x^3-4x^2+x+6$?

    $0$. $f(2) = 8-16+2+6 = 0$, confirming $(x-2)$ is a factor.

61. What are the solutions of $x^2=49$?

    $x=7$ or $x=-7$. Take square root of both sides: $x=\pm\sqrt{49}$.

62. Factor completely: $x^2-16$.

    $(x-4)(x+4)$. Difference of squares: $x^2-4^2=(x-4)(x+4)$.

63. What is the difference of squares factoring pattern?

    $a^2-b^2=(a-b)(a+b)$. Factors any expression of form $a^2-b^2$.

64. What are the solutions of $x^2-5x+6=0$?

    $x=2$ or $x=3$. Factor as $(x-2)(x-3)=0$ or use quadratic formula.

65. Which value is a root of $f(x)=x^3-4x^2-x+4$: $x=1$ or $x=2$?

    $x=1$. $f(1)=1-4-1+4=0$, so $x=1$ is a root.

66. State the quadratic formula for solving $ax^2+bx+c=0$.

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Solves $ax^2+bx+c=0$ when factoring is difficult.

67. What is the definition of a polynomial equation in $x$?

    An equation setting a polynomial in $x$ equal to $0$. A polynomial expression set equal to zero forms a polynomial equation.

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

    $0$. Discriminant $= 16-20 = -4 < 0$, so no real solutions exist.

69. What is the leading coefficient of $-4x^3+2x-1$?

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

70. Factor the trinomial $x^2+7x+12$.

    $(x+3)(x+4)$. Find two numbers that multiply to $12$ and add to $7$: $3$ and $4$.

71. What is the degree of the polynomial $7x^4-3x^2+x-9$?

    $4$. The degree is the highest exponent of the variable.

72. Solve $x^2-9=0$.

    $x=3$ or $x=-3$. Factor as $(x-3)(x+3)=0$, then apply zero product property.

73. What is the result of factoring $x^2+7x+12$ completely?

    $(x+3)(x+4)$. Find two numbers that multiply to $12$ and add to $7$: $3$ and $4$.

74. What is the constant term of $3x^4-x+8$?

    $8$. The constant term is the term without any variable.

75. What is the maximum possible number of real solutions to a degree $n$ polynomial equation?

    At most $n$ real solutions. Fundamental Theorem: degree $n$ polynomial has at most $n$ roots.

76. Solve the polynomial equation $(x-4)(x+1)=0$.

    $x=4$ or $x=-1$. Set each factor to zero: $x-4=0$ or $x+1=0$.

77. What is the leading coefficient of $-5x^3+2x-1$?

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

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

    $x=1$ or $x=5$. Factor as $(x-1)(x-5)=0$, then apply zero-product property.

79. What is the factorization pattern for a difference of squares $a^2-b^2$?

    $a^2-b^2=(a-b)(a+b)$. Difference of squares factors into sum times difference.

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

    The remainder equals $f(r)$. Substitute $r$ into $f(x)$ to get remainder.

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

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

82. What is the perfect square trinomial pattern for $a^2+2ab+b^2$?

    $(a+b)^2$. Recognizes when a trinomial is a perfect square.

83. Solve for $x$: $x^2=16$.

    $x=\pm^4$. Take square root of both sides: $x=\pm\sqrt{16}=\pm^4$.

84. State the quadratic formula for solving $ax^2+bx+c=0$.

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Formula solves any quadratic equation.

85. What is the factored form of the difference of squares $a^2-b^2$?

    $(a-b)(a+b)$. Difference of squares factors as product of sum and difference.

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

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. General formula for solving any quadratic equation.

87. What is the discriminant of $ax^2+bx+c=0$, and what does it determine?

    Discriminant $b^2-4ac$; it determines the number of real solutions. Positive gives 2 real roots, zero gives 1, negative gives 0.

88. What is the sum of the solutions of $x^2-6x+11=0$ using $-\frac{b}{a}$?

    $6$. For $ax^2+bx+c=0$, sum of roots equals $-\frac{b}{a} = -\frac{-6}{1} = 6$.

89. What is the discriminant of $2x^2-3x+5=0$?

    $-31$. Calculate $b^2-4ac = (-3)^2-4(2)(5) = 9-40$.

90. What does the Zero Product Property state for $ab=0$?

    If $ab=0$, then $a=0$ or $b=0$. At least one factor must be zero.

91. State the zero-product property for $ab=0$.

    If $ab=0$, then $a=0$ or $b=0$. A product equals zero only if at least one factor equals zero.

92. Identify the solutions of $x^2-5x+6=0$.

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

93. What is the complete factorization of $x^2-49$ over the integers?

    $(x-7)(x+7)$. $49 = 7^2$, so apply difference of squares pattern.

94. Factor completely: $x^2+6x+9$.

    $(x+3)^2$. Recognizes as $(x)^2+2(x)(3)+(3)^2$, a perfect square trinomial.

95. What is the degree of the polynomial $7x^4-3x^2+9$?

    $4$. The degree is the highest power of the variable.

96. What is the product of the solutions of $2x^2+3x-5=0$ using $\frac{c}{a}$?

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

97. What does a discriminant value $b^2-4ac<0$ imply about real solutions?

    No real solutions. Negative discriminant means no real roots exist.

98. Identify the number of real solutions to $x^2+4x+5=0$.

    $0$ real solutions. Discriminant $16-20=-4<0$.

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

    $b^2-4ac$. Determines the nature and number of roots for quadratics.

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

     $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Solves any quadratic equation when $a\neq 0$.

101. Identify whether $(x-2)$ is a factor of $f(x)=x^3-4x^2+4x$.

     Yes, $(x-2)$ is a factor. $f(2)=8-16+8=0$, so $(x-2)$ is a factor by Factor Theorem.

102. Identify the solutions of $x^3-27=0$.

     $x=3$. $x^3=27$, so $x=\sqrt[3]{27}=3$.

103. What is the result of factoring $x^2-5x-14$ completely?

     $(x-7)(x+2)$. Find two numbers that multiply to $-14$ and add to $-5$: $-7$ and $2$.

104. State the Zero Product Property for equations like $(x-2)(x+5)=0$.

     If $ab=0$, then $a=0$ or $b=0$. A product equals zero only if at least one factor is zero.

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

     $0$ real solutions. Discriminant $=16-20=-4<0$, so no real solutions exist.

106. What is the factored form of the perfect square trinomial $a^2+2ab+b^2$?

     $(a+b)^2$. Perfect square with positive middle term factors as $(a+b)$ squared.

107. What is the definition of a polynomial equation in one variable $x$?

     An equation setting a polynomial in $x$ equal to $0$. Sets polynomial to zero to find roots.

108. What is the leading coefficient of $-5x^3+2x-8$?

     Leading coefficient $-5$. The coefficient of the highest degree term.

109. What does the discriminant value $b^2-4ac=0$ imply about real solutions?

     One real solution (a repeated root). Zero discriminant produces one repeated real root.

110. Identify the real solutions of $x^3-4x=0$.

     $x=-2,0,2$. Factor out $x$: $x(x^2-4) = x(x-2)(x+2) = 0$.

111. Factor $x^2-16$ completely.

     $(x-4)(x+4)$. Apply difference of squares: $x^2-16=x^2-4^2=(x-4)(x+4)$.

112. What is the degree of the polynomial $7x^4-3x^2+9$?

     $4$. The degree is the highest power of $x$ in the polynomial.

113. Identify the solutions of $(x-4)(x+1)=0$.

     $x=4$ and $x=-1$. Apply zero product property.

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

     $-31$. Use $b^2-4ac$ with $a=2$, $b=-3$, $c=5$: $(-3)^2-4(2)(5)=9-40=-31$.

115. What is the remainder when $f(x)=x^3-2x+5$ is divided by $(x-2)$?

     $f(2)=9$. By Remainder Theorem, substitute $x=2$.

116. What are the solutions of $x^2+8x=0$?

     $x=0$ or $x=-8$. Factor out $x$: $x(x+8)=0$.

117. What are the solutions of $(x-4)(x+1)=0$?

     $x=4$ or $x=-1$. Apply Zero Product Property: set each factor to zero.

118. Factor $x^2+10x+25$ completely.

     $(x+5)^2$. Perfect square: $x^2+10x+25=x^2+2(5)x+5^2=(x+5)^2$.

119. Which value makes $x^2-3x-10$ equal to $0$: $x=5$ or $x=-5$?

     $x=5$. Check: $25-15-10=0$ ✓; $25+15-10=30 \neq 0$.

120. Identify the solutions of $2x^2-8x=0$.

     $x=0$ and $x=4$. Factor out $2x$: $2x(x-4)=0$.

121. Solve for $x$: $(x-4)(x+1)=0$.

     $x=4$ or $x=-1$. Apply zero-product property to each factor.

122. What is the degree of a polynomial, in terms of its exponents?

     The greatest exponent of the variable with a nonzero coefficient. Highest power determines polynomial's degree.

123. What is the vertex form of a quadratic polynomial?

     $a(x-h)^2+k$. Shows vertex at $(h,k)$ directly.

124. Factor the difference of squares $x^2-16$.

     $(x-4)(x+4)$. Uses pattern $a^2-b^2=(a-b)(a+b)$ with $a=x$, $b=4$.

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

     $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Solves any quadratic equation in standard form.

126. Factor the perfect square trinomial $x^2+10x+25$.

     $(x+5)^2$. Recognizes pattern $a^2+2ab+b^2=(a+b)^2$ with $a=x$, $b=5$.

127. What is the Factor Theorem statement relating $f(r)$ to the factor $(x-r)$?

     $(x-r)$ is a factor of $f(x)$ iff $f(r)=0$. A polynomial has factor $(x-r)$ exactly when $r$ is a root.

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