AP Precalculus Flashcards: Equivalent Polynomial And Rational Expressions

What this deck covers

This deck focuses on Equivalent Polynomial And Rational Expressions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

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 is the standard form of a quadratic polynomial?

Answer: $ax^2 + bx + c$. The general form where $a \neq 0$.

Flashcard 2: What is the quotient of $\frac{x^2 - 4}{x - 2}$?

Answer: $x + 2$. Factor numerator: $\frac{(x+2)(x-2)}{x-2} = x+2$.

Flashcard 3: What is the expression for $(x - 3)^2$?

Answer: $x^2 - 6x + 9$. Use $(a-b)^2 = a^2 - 2ab + b^2$.

Flashcard 4: What is the sum of the roots of $x^2 - 4x + 4$?

Answer:

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

Flashcard 5: Rewrite $\frac{x^2 + 5x + 6}{x + 2}$ in simplest form.

Answer: $x + 3$. Factor and cancel: $\frac{(x+2)(x+3)}{x+2} = x+3$.

Flashcard 6: Which term refers to the highest degree term in a polynomial?

Answer: Leading term. The term with the highest degree in standard form.

Flashcard 7: Identify the expression equivalent to $\frac{x^3 - x}{x}$.

Answer: $x^2 - 1$. Factor and cancel: $\frac{x(x^2-1)}{x} = x^2-1$.

Flashcard 8: Simplify $\frac{x^2 - 3x}{x - 3}$.

Answer: $x$. Factor numerator: $\frac{x(x-3)}{x-3} = x$.

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

Answer:

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

Flashcard 10: What is the standard form of a linear polynomial?

Answer: $ax + b$. The general form of a first-degree polynomial.

Flashcard 11: Find the product of $(x + 1)$ and $(x - 1)$.

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

Flashcard 12: Identify the expression equivalent to $\frac{x^2 + 2x}{x}$.

Answer: $x + 2$. Factor out $x$: $\frac{x(x+2)}{x} = x+2$.

Flashcard 13: State the discriminant of the quadratic $x^2 + 4x + 4$.

Answer:

1. For perfect square, discriminant $b^2 - 4ac = 0$.

Flashcard 14: Convert $\frac{3x + 6}{x + 2}$ to simplest form.

Answer:

1. Factor and cancel: $\frac{3(x+2)}{x+2} = 3$.

Flashcard 15: What is the expression for the cube of $(x + 2)$?

Answer: $x^3 + 6x^2 + 12x + 8$. Use $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

Flashcard 16: Simplify the expression $\frac{x^2 - 4x}{x}$.

Answer: $x - 4$. Factor out $x$: $\frac{x(x-4)}{x} = x-4$.

Flashcard 17: What is the binomial theorem?

Answer: $(a + b)^n = \text{sum of terms } C(n, k)a^{n-k}b^k$. Expands $(a + b)^n$ using combinations and powers.

Flashcard 18: 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 19: Identify the leading coefficient in $5x^4 - 3x^2 + x$.

Answer:

1. The coefficient of the highest degree term.

Flashcard 20: What is the expression for the sum of $2x^2 + 3x$ and $x^2 - x$?

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

Flashcard 21: Simplify the expression $2x(x^2 - 3x + 4)$.

Answer: $2x^3 - 6x^2 + 8x$. Distribute $2x$ to each term inside parentheses.

Flashcard 22: What does it mean for a polynomial to be monic?

Answer: Leading coefficient is 1. The coefficient of the highest degree term equals 1.

Flashcard 23: Factor $x^2 + 6x + 9$.

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

Flashcard 24: What is the expression for the difference between $x^2 + 4x$ and $2x^2 + x$?

Answer: $-x^2 + 3x$. $(x^2 + 4x) - (2x^2 + x) = -x^2 + 3x$.

Flashcard 25: State the expression for the difference of cubes $a^3 - b^3$.

Answer: $(a - b)(a^2 + ab + b^2)$. The factorization formula for $a^3 - b^3$.

Flashcard 26: What is the polynomial remainder theorem?

Answer: Remainder of $f(x)$ divided by $x - a$ is $f(a)$. Substitute $x=a$ into $f(x)$ to find remainder.

Flashcard 27: Express $3x^2 - 12$ as a product of its factors.

Answer: $3(x^2 - 4)$. Factor out common factor 3 first.

Flashcard 28: State the remainder when $x^3 + 3x^2 - x + 5$ is divided by $x - 1$.

Answer:

1. By remainder theorem, substitute $x=1$: $1+3-1+5=8$.

Flashcard 29: What is the product of the roots of $x^2 - 5x + 6$?

Answer:

1. For $ax^2 + bx + c$, product of roots is $\frac{c}{a}$.

Flashcard 30: What is the term for a polynomial with three terms?

Answer: Trinomial. A polynomial with exactly three terms.

Flashcard 31: What is the expression for the square of $(x + 2)$?

Answer: $x^2 + 4x + 4$. Use $(a+b)^2 = a^2 + 2ab + b^2$.

Flashcard 32: Simplify the expression $\frac{x^2 - 9}{x + 3}$.

Answer: $x - 3$. Factor numerator: $\frac{(x+3)(x-3)}{x+3} = x-3$.

Flashcard 33: What is the greatest common factor of $6x^2 + 9x$?

Answer: 3x. Factor out the GCF: $3x(2x + 3)$.

Flashcard 34: Express $x^3 - 27$ as a product of factors.

Answer: $(x - 3)(x^2 + 3x + 9)$. Difference of cubes: $a^3 - b^3 = (a-b)(a^2+ab+b^2)$.

Flashcard 35: What is the common denominator of $\frac{1}{x}$ and $\frac{1}{x^2}$?

Answer: $x^2$. Use the highest power of $x$ in both denominators.

Flashcard 36: What is the expanded form of $(x + 1)^3$?

Answer: $x^3 + 3x^2 + 3x + 1$. Use $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

Flashcard 37: State the zeroes of the polynomial $x^2 - 5x + 6$.

Answer: 2, 3. Factor as $(x-2)(x-3)$ and set each factor to zero.

Flashcard 38: Convert $f(x) = (x - 2)(x + 3)$ to standard form.

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