ACT Math Flashcards: Quadratics And Polynomials

What this deck covers

This deck focuses on Quadratics And Polynomials, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

Answer:

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

Flashcard 4: For what discriminant values does $ax^2 + bx + c = 0$ have real and equal roots?

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

Flashcard 5: Find $f(2)$ for $f(x) = x^2 - 3x + 2$.

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

Flashcard 6: Factor $x^2 - 1$.

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

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

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

Flashcard 8: State the difference of squares factorization formula.

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

Flashcard 9: State the factor theorem relating $f(a)$ to the factor $(x-a)$.

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

Flashcard 10: What are the roots of $x^2 - 9$?

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

Flashcard 11: What does $\Delta=0$ indicate about the real solutions of $ax^2+bx+c=0$?

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

Flashcard 12: What is the vertex form of a quadratic function?

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

Flashcard 13: State the binomial theorem.

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

Flashcard 14: Factor $x^2 - 9$ completely.

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

Flashcard 15: What is the result of multiplying $x(x+1)$?

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

Flashcard 16: Find the discriminant of $ax^2 + bx + c = 0$.

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

Flashcard 17: Identify the axis of symmetry for $y = ax^2 + bx + c$.

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

Flashcard 18: State the difference of squares factorization formula.

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

Flashcard 19: What is the $y$-intercept of $y=ax^2+bx+c$?

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

Flashcard 20: State the definition of the degree of a polynomial.

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

Flashcard 21: Simplify $2x^2 - 4x + 2$. What is the expression?

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

Flashcard 22: State the quadratic formula for solutions to $ax^2+bx+c=0$.

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

Flashcard 23: Which term is the quadratic term in $5x^2 - 3x + 8$?

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

Flashcard 24: What is the standard form of a quadratic equation?

Answer: $ax^2 + bx + c = 0$. General form with $a \neq 0$ for quadratic equations.

Flashcard 25: State Vieta's formulas for $ax^2+bx+c=0$ with roots $r_1,r_2$.

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

Flashcard 26: Evaluate $x^2 - 4x + 4$ for $x = 2$.

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

Flashcard 27: Solve for $x$: $x^2 - 5x + 6 = 0$

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

Flashcard 28: What is the degree of the polynomial $7x^5 - 4x^3 + 2$?

Answer:

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

Flashcard 29: For what discriminant values does $ax^2 + bx + c = 0$ have complex roots?

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

Flashcard 30: Identify the constant term in $3x^2 - 5x + 7$.

Answer:

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

Flashcard 31: Simplify $(x + 1)(x - 1)$. What is the expression?

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

Flashcard 32: What is the y-intercept of $y = 3x^2 + 2x - 4$?

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

Flashcard 33: Factor $x^2 - 4x + 4$.

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

Flashcard 34: What is the vertex of $y = (x-2)^2 + 3$?

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

Flashcard 35: State the zero-product property used when solving $(A)(B)=0$.

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

Flashcard 36: Find $f(2)$ for $f(x) = x^2 - 3x + 2$.

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

Flashcard 37: Find the discriminant of $3x^2-4x+5=0$.

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

Flashcard 38: For what discriminant values does $ax^2 + bx + c = 0$ have real and distinct roots?

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

Flashcard 39: Factor $x^2 - 4x + 4$.

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

Flashcard 40: Find the vertex of $y = x^2 - 4x + 4$.

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

Flashcard 41: State the standard form of a quadratic function in $x$.

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

Flashcard 42: Find the remainder when $x^3 - 2x^2 + x - 1$ is divided by $x - 1$.

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

Flashcard 43: What is the discriminant of $ax^2+bx+c=0$?

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

Flashcard 44: Find the greatest common factor of $2x^3$ and $4x^2$.

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

Flashcard 45: What are the roots of $x^2 - 9$?

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

Flashcard 46: If $f(x)=x^2-1$, what are the zeros of $f(x)$?

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

Flashcard 47: State the factor theorem.

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

Flashcard 48: Identify the vertex $x$-coordinate of $y=2x^2-8x+1$.

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

Flashcard 49: Subtract $(3x^2 + 2x)$ from $(5x^2 + x + 4)$.

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

Flashcard 50: What does $\Delta=0$ indicate about the real solutions of $ax^2+bx+c=0$?

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

Flashcard 51: Evaluate $x^2 - 4x + 4$ for $x = 2$.

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

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

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

Flashcard 53: Simplify $2x^2 - 4x + 2$. What is the expression?

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

Flashcard 54: What is the linear term in the polynomial $7x^2 + 2x - 1$?

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

Flashcard 55: State the remainder theorem for dividing $f(x)$ by $(x-a)$.

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

Flashcard 56: State the axis of symmetry for $y=ax^2+bx+c$.

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

Flashcard 57: What does the discriminant indicate about the roots?

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

Flashcard 58: Identify the vertex form of a quadratic function.

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

Flashcard 59: In $y=a(x-h)^2+k$, what are the vertex coordinates?

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

Flashcard 60: Find the remainder when $x^3 - 2x^2 + x - 1$ is divided by $x - 1$.

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

Flashcard 61: Simplify $x^3 - x^2 + x - 1$ for $x = 1$.

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

Flashcard 62: Determine the roots of $x^2 - 5x + 6 = 0$.

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

Flashcard 63: What is the standard form of a quadratic equation?

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

Flashcard 64: How many real solutions does $x^2+4x+5=0$ have?

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

Flashcard 65: Simplify: $(x+2)(x-5)$.

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

Flashcard 66: What is the product of $(x+2)$ and $(x-3)$?

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

Flashcard 67: State the factored form of a quadratic with roots $r_1$ and $r_2$.

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

Flashcard 68: State Vieta's formulas for $ax^2+bx+c=0$ with roots $r_1,r_2$.

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

Flashcard 69: What is the constant term in the polynomial $x^3 + 4x^2 - 2x + 6$?

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

Flashcard 70: What is the discriminant of $ax^2+bx+c=0$?

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

Flashcard 71: Solve for $x$: $x^2 - 5x + 6 = 0$

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

Flashcard 72: Determine if $x = -2$ is a root of $x^3 + 2x^2 - 5$.

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

Flashcard 73: What does $\Delta<0$ indicate about the solutions of $ax^2+bx+c=0$?

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

Flashcard 74: What is the y-intercept of $y = 3x^2 + 2x - 4$?

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

Flashcard 75: What is the result of multiplying $x(x+1)$?

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

Flashcard 76: What is the sum of the coefficients in $2x^2 - 3x + 5$?

Answer:

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

Flashcard 77: Factor completely: $x^2+10x+25$.

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

Flashcard 78: State the zero-product property used when solving $(A)(B)=0$.

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

Flashcard 79: Evaluate $f(2)$ for $f(x)=x^3-4x+1$.

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

Flashcard 80: Find the vertex of $y=x^2-6x+5$.

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

Flashcard 81: State the maximum number of real zeros a degree $n$ polynomial can have.

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

Flashcard 82: What is the degree of the polynomial $7x^5 - 4x^3 + 2$?

Answer:

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

Flashcard 83: Identify the cubic term in $4x^3 + 3x^2 + x + 5$.

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

Flashcard 84: Expand and simplify: $(x-4)^2$.

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

Flashcard 85: Find the vertex of $y=x^2-6x+5$.

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

Flashcard 86: Identify the degree of $x^3 + 2x^2 - 5x + 1$.

Answer:

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

Flashcard 87: State the factored form of a quadratic with roots $r_1$ and $r_2$.

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

Flashcard 88: What is the coefficient of $x$ in $5x^2 + 3x - 7$?

Answer:

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

Flashcard 89: Evaluate $f(2)$ for $f(x)=x^3-4x+1$.

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

Flashcard 90: What is the vertex of $y = (x-2)^2 + 3$?

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

Flashcard 91: What is the sum of the solutions to $2x^2 - 8x = 0$?

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

Flashcard 92: Determine if $x = -2$ is a root of $x^3 + 2x^2 - 5$.

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

Flashcard 93: For what discriminant values does $ax^2 + bx + c = 0$ have real and equal roots?

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

Flashcard 94: Simplify $(2x - 3)^2$. What is the expression?

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

Flashcard 95: Determine the x-intercepts of $y = x^2 - 4$.

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

Flashcard 96: State the quadratic formula.

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

Flashcard 97: What is the product of the roots of $ax^2 + bx + c = 0$?

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

Flashcard 98: Factor completely: $x^2-9$.

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

Flashcard 99: Find the zero of the polynomial $x - 4$.

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

Flashcard 100: What is the sum of the roots of $ax^2 + bx + c = 0$?

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