SAT Math Flashcards: Systems Of Polynomial Equations

What this deck covers

This deck focuses on Systems Of Polynomial Equations, giving you a quick way to review the definitions, rules, and examples that matter most for SAT 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: What method can be used to solve a system of polynomial equations graphically?

Answer: Graph the equations and find intersection points. Solutions occur where the curves visually cross on the coordinate plane.

Flashcard 2: Solve: $y = x^2 - 4$ and $y = 0$.

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

Flashcard 3: What method can be used to solve a system of polynomial equations graphically?

Answer: Graph the equations and find intersection points. Solutions occur where the curves visually cross on the coordinate plane.

Flashcard 4: Solve: $x^2 + y^2 = 25$ and $x = 3y$.

Answer: $(x, y) = (3, 1)$ or $(x, y) = (-3, -1)$. Substitute $x = 3y$ into circle equation.

Flashcard 5: Find the $x$-intercepts of $y = x^2 - 16$.

Answer: $x = 4$ or $x = -4$. Set $y = 0$ and solve $x^2 - 16 = 0$.

Flashcard 6: What is the factored form of $x^2 + 5x + 6$?

Answer: $(x + 2)(x + 3)$. Find two numbers that multiply to 6 and add to 5.

Flashcard 7: Identify the roots of $x^2 - 4x = 0$.

Answer: $x = 0$ or $x = 4$. Factor out $x$: $x(x - 4) = 0$.

Flashcard 8: Convert $x^2 - 4x + 4$ to vertex form.

Answer: $(x - 2)^2$. Perfect square trinomial in factored form.

Flashcard 9: Solve for $y$: $y^2 - 4y + 4 = 0$.

Answer: $y = 2$. Perfect square trinomial with double root.

Flashcard 10: Identify the solution of the system: $x^2 + y = 5$ and $x + y^2 = 5$.

Answer: $(1, 2)$ and $(2, 1)$. Both points satisfy both equations when substituted.

Flashcard 11: Identify the constant term in $3x^3 - 4x + 5$.

Answer:

1. Term without any variable.

Flashcard 12: What is the solution to $x^2 + 6x + 9 = 0$?

Answer: $x = -3$. Perfect square trinomial $(x + 3)^2 = 0$.

Flashcard 13: What is the solution to $x^2 + 6x + 9 = 0$?

Answer: $x = -3$. Perfect square trinomial $(x + 3)^2 = 0$.

Flashcard 14: Solve: $x^2 - 2x - 8 = 0$.

Answer: $x = 4$ or $x = -2$. Factor as $(x - 4)(x + 2) = 0$.

Flashcard 15: What is the vertex of $y = (x - 1)^2 - 4$?

Answer: $(1, -4)$. Vertex form shows vertex at $(h, k)$.

Flashcard 16: What is the $y$-intercept of $y = 3x + 4$?

Answer:

1. Value where line crosses the $y$-axis.

Flashcard 17: What is the solution to the system $x + y = 5$ and $x - y = 1$?

Answer: $(x, y) = (3, 2)$. Add equations to get $2x = 4$, then substitute back.

Flashcard 18: What is the factored form of $x^2 - 9$?

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

Flashcard 19: What is the degree of the polynomial $5x^4 - 3x^2 + 2$?

Answer:

1. Highest power of the variable term.

Flashcard 20: What is the factored form of $x^2 - 2x - 3$?

Answer: $(x - 3)(x + 1)$. Factor by finding two numbers that multiply to $-3$.

Flashcard 21: What is the definition of a polynomial?

Answer: An expression of finite terms with non-negative integer exponents. Sum of terms with variables raised to whole number powers.

Flashcard 22: What is the factored form of $x^2 - 2x - 3$?

Answer: $(x - 3)(x + 1)$. Factor by finding two numbers that multiply to $-3$.

Flashcard 23: Convert $x^2 - 4x + 4$ to vertex form.

Answer: $(x - 2)^2$. Perfect square trinomial in factored form.

Flashcard 24: What is the solution to $y = 3x - 2$ and $y = -x + 6$?

Answer: $(x, y) = (2, 4)$. Set equations equal: $3x - 2 = -x + 6$.

Flashcard 25: What is the formula for the discriminant of a quadratic equation $ax^2 + bx + c = 0$?

Answer: $b^2 - 4ac$. Determines nature of roots for quadratic equations.

Flashcard 26: Solve for $x$: $x^2 + y^2 = 13$ and $x + y = 3$.

Answer: $x = 2$ or $x = 1$. Substitute $y = 3 - x$ into first equation and solve.

Flashcard 27: What is the degree of $4x^5 - x^3 + 2x - 1$?

Answer:

1. Highest power determines polynomial degree.

Flashcard 28: What is the sum of the roots of $ax^2 + bx + c = 0$?

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

Flashcard 29: What is the general form of a system of polynomial equations?

Answer: $f(x, y) = 0, \, g(x, y) = 0$. Two polynomial equations set equal to zero that must be solved simultaneously.

Flashcard 30: What is the solution to the system $x + y = 5$ and $x - y = 1$?

Answer: $(x, y) = (3, 2)$. Add equations to get $2x = 4$, then substitute back.

Flashcard 31: Find the vertex form of $y = x^2 - 4x + 4$.

Answer: $y = (x - 2)^2$. Complete the square to get vertex form.

Flashcard 32: Solve: $y = 2x + 3$ and $y = -x + 1$.

Answer: $(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equations equal: $2x + 3 = -x + 1$.

Flashcard 33: Find the $x$-intercepts of $y = x^2 - 16$.

Answer: $x = 4$ or $x = -4$. Set $y = 0$ and solve $x^2 - 16 = 0$.

Flashcard 34: What are the solutions to $x^2 = 9$?

Answer: $x = 3$ or $x = -3$. Take square root of both sides.

Flashcard 35: Solve: $x^2 + y^2 = 25$ and $x = 3y$.

Answer: $(x, y) = (3, 1)$ or $(x, y) = (-3, -1)$. Substitute $x = 3y$ into circle equation.

Flashcard 36: What is the factored form of $x^2 - 9$?

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

Flashcard 37: Is the system $x^2 + y^2 = 1$ and $x + y = 1$ linear or non-linear?

Answer: Non-linear. The presence of $x^2 + y^2$ makes the system non-linear.

Flashcard 38: What is the $y$-intercept of $y = 3x + 4$?

Answer: $4$. Value where line crosses the $y$-axis.

Flashcard 39: Is the system $x^2 + y^2 = 1$ and $x + y = 1$ linear or non-linear?

Answer: Non-linear. The presence of $x^2 + y^2$ makes the system non-linear.

Flashcard 40: Identify the leading coefficient of $3x^4 - x^3 + 2x^2$.

Answer:

1. Coefficient of the highest degree term.

Flashcard 41: Identify the leading coefficient of $3x^4 - x^3 + 2x^2$.

Answer:

1. Coefficient of the highest degree term.

Flashcard 42: What is the formula for the discriminant of a quadratic equation $ax^2 + bx + c = 0$?

Answer: $b^2 - 4ac$. Determines nature of roots for quadratic equations.

Flashcard 43: What is the vertex of $y = x^2 + 6x + 9$?

Answer: $(-3, 0)$. Perfect square $(x + 3)^2$ has vertex at $x = -3$.

Flashcard 44: Solve the system: $2x + 3y = 7$ and $4x - y = 1$.

Answer: $(x, y) = (1, \frac{5}{3})$. Solve by elimination or substitution method.

Flashcard 45: What is the solution to $x^2 + 1 = 0$?

Answer: $x = i$ or $x = -i$. Complex roots when discriminant is negative.

Flashcard 46: Solve: $x^2 - 2x - 8 = 0$.

Answer: $x = 4$ or $x = -2$. Factor as $(x - 4)(x + 2) = 0$.

Flashcard 47: What is the sum of the roots of $ax^2 + bx + c = 0$?

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

Flashcard 48: What is the solution to $x^2 + 2x + 1 = 0$?

Answer: $x = -1$. Perfect square trinomial $(x + 1)^2 = 0$.

Flashcard 49: Solve for $x$: $x^2 + y^2 = 13$ and $x + y = 3$.

Answer: $x = 2$ or $x = 1$. Substitute $y = 3 - x$ into first equation and solve.

Flashcard 50: Solve: $y = x^2 - 4$ and $y = 0$.

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

Flashcard 51: Identify the roots of $x^2 - 4x = 0$.

Answer: $x = 0$ or $x = 4$. Factor out $x$: $x(x - 4) = 0$.

Flashcard 52: Solve: $x^2 - 5x + 6 = 0$.

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

Flashcard 53: Find a common solution to $x^2 - 4x + 4 = 0$ and $y = x - 2$.

Answer: $(2, 0)$. First equation gives $x = 2$, then substitute into second equation.

Flashcard 54: What is the definition of a polynomial?

Answer: An expression of finite terms with non-negative integer exponents. Sum of terms with variables raised to whole number powers.

Flashcard 55: What is the quadratic formula?

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

Flashcard 56: Solve for $x$: $x^2 + 4x + 4 = 0$.

Answer: $x = -2$. Perfect square trinomial $(x + 2)^2 = 0$.

Flashcard 57: What is the general form of a quadratic equation?

Answer: $ax^2 + bx + c = 0$. Standard form with $a \neq 0$.

Flashcard 58: Solve: $x^2 - 5x + 6 = 0$.

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

Flashcard 59: What is the degree of a system consisting of $x^3 + y^3 = 7$ and $xy = 1$?

Answer: Degree 3. The highest degree among all terms in the system is 3.

Flashcard 60: What is the degree of the polynomial $5x^4 - 3x^2 + 2$?

Answer:

1. Highest power of the variable term.

Flashcard 61: What is the leading term of $5x^3 + 4x^2 - x$?

Answer: $5x^3$. Term with highest degree in the polynomial.

Flashcard 62: What is the general form of a quadratic equation?

Answer: $ax^2 + bx + c = 0$. Standard form with $a \neq 0$.

Flashcard 63: What do the solutions of a system of polynomial equations represent?

Answer: Intersection points of the graphs. Each solution represents a point where all equations are satisfied.

Flashcard 64: What are the solutions to $x^2 = 9$?

Answer: $x = 3$ or $x = -3$. Take square root of both sides.

Flashcard 65: What is the product of the roots of $ax^2 + bx + c = 0$?

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

Flashcard 66: What is the degree of the polynomial $2x^3 - 4x^2 + x - 5$?

Answer:

1. Highest power of the variable in the polynomial.

Flashcard 67: What is the leading term of $5x^3 + 4x^2 - x$?

Answer: $5x^3$. Term with highest degree in the polynomial.

Flashcard 68: What is the degree of the polynomial $2x^3 - 4x^2 + x - 5$?

Answer:

1. Highest power of the variable in the polynomial.

Flashcard 69: What is the solution to $y = 3x - 2$ and $y = -x + 6$?

Answer: $(x, y) = (2, 4)$. Set equations equal: $3x - 2 = -x + 6$.

Flashcard 70: What is the degree of a system consisting of $x^3 + y^3 = 7$ and $xy = 1$?

Answer: Degree 3. The highest degree among all terms in the system is 3.

Flashcard 71: Solve the system: $2x + 3y = 7$ and $4x - y = 1$.

Answer: $(x, y) = (1, \frac{5}{3})$. Solve by elimination or substitution method.

Flashcard 72: What is the solution to $x^2 + 1 = 0$?

Answer: $x = i$ or $x = -i$. Complex roots when discriminant is negative.

Flashcard 73: Solve for $x$: $x^2 + 4x + 4 = 0$.

Answer: $x = -2$. Perfect square trinomial $(x + 2)^2 = 0$.

Flashcard 74: Find a common solution to $x^2 - 4x + 4 = 0$ and $y = x - 2$.

Answer: $(2, 0)$. First equation gives $x = 2$, then substitute into second equation.

Flashcard 75: What is the degree of $4x^5 - x^3 + 2x - 1$?

Answer: $5$. Highest power determines polynomial degree.

Flashcard 76: Solve: $y = 2x + 3$ and $y = -x + 1$.

Answer: $(x, y) = (-\frac{2}{3}, \frac{5}{3})$. Set equations equal: $2x + 3 = -x + 1$.

Flashcard 77: Find the $y$-intercept of $y = x^2 + 3x + 2$.

Answer:

1. Substitute $x = 0$ into the equation.

Flashcard 78: Which method of solving is suitable for non-linear systems?

Answer: Substitution or elimination. Both methods work by reducing the system to simpler equations.

Flashcard 79: Find the $y$-intercept of $y = x^2 + 3x + 2$.

Answer:

1. Substitute $x = 0$ into the equation.

Flashcard 80: What is the general form of a system of polynomial equations?

Answer: $f(x, y) = 0, \, g(x, y) = 0$. Two polynomial equations set equal to zero that must be solved simultaneously.

Flashcard 81: Solve the system: $x^2 + y^2 = 25$ and $y = 3x$. Find one solution.

Answer: $(3, 9)$. Substitute $y = 3x$ into the first equation to find $x = 3$.

Flashcard 82: Solve the system: $x^2 + y^2 = 25$ and $y = 3x$. Find one solution.

Answer: $(3, 9)$. Substitute $y = 3x$ into the first equation to find $x = 3$.

Flashcard 83: What is the solution to $x^2 + 2x + 1 = 0$?

Answer: $x = -1$. Perfect square trinomial $(x + 1)^2 = 0$.

Flashcard 84: What is the quadratic formula?

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

Flashcard 85: Identify the constant term in $3x^3 - 4x + 5$.

Answer:

1. Term without any variable.

Flashcard 86: Find the vertex form of $y = x^2 - 4x + 4$.

Answer: $y = (x - 2)^2$. Complete the square to get vertex form.

Flashcard 87: What is the solution to $y = x^2 - 1$ and $y = 0$?

Answer: $x = 1$ or $x = -1$. Set $x^2 - 1 = 0$ and factor.

Flashcard 88: What do the solutions of a system of polynomial equations represent?

Answer: Intersection points of the graphs. Each solution represents a point where all equations are satisfied.

Flashcard 89: What is the factored form of $x^2 + 5x + 6$?

Answer: $(x + 2)(x + 3)$. Find two numbers that multiply to 6 and add to 5.

Flashcard 90: What is the vertex of $y = (x - 1)^2 - 4$?

Answer: $(1, -4)$. Vertex form shows vertex at $(h, k)$.

Flashcard 91: Which method of solving is suitable for non-linear systems?

Answer: Substitution or elimination. Both methods work by reducing the system to simpler equations.

Flashcard 92: What is the product of the roots of $ax^2 + bx + c = 0$?

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

Flashcard 93: Solve for $y$: $y^2 - 4y + 4 = 0$.

Answer: $y = 2$. Perfect square trinomial with double root.

Flashcard 94: What is the vertex of $y = x^2 + 6x + 9$?

Answer: $(-3, 0)$. Perfect square $(x + 3)^2$ has vertex at $x = -3$.

Flashcard 95: What is the solution to $y = x^2 - 1$ and $y = 0$?

Answer: $x = 1$ or $x = -1$. Set $x^2 - 1 = 0$ and factor.