PSAT Math Flashcards: Polynomial Equations

What this deck covers

This deck focuses on Polynomial Equations, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT 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 does the discriminant value $b^2-4ac>0$ imply about real solutions?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: 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 $n$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 63: What is the difference of squares factoring pattern?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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