Equivalent Polynomial and Rational Expressions - AP Precalculus
Card 1 of 30
Simplify $\frac{x^2 - 3x}{x - 3}$.
Simplify $\frac{x^2 - 3x}{x - 3}$.
Tap to reveal answer
$x$. Factor numerator: $\frac{x(x-3)}{x-3} = x$.
$x$. Factor numerator: $\frac{x(x-3)}{x-3} = x$.
← Didn't Know|Knew It →
What is the standard form of a quadratic polynomial?
What is the standard form of a quadratic polynomial?
Tap to reveal answer
$ax^2 + bx + c$. The general form where $a \neq 0$.
$ax^2 + bx + c$. The general form where $a \neq 0$.
← Didn't Know|Knew It →
What is the expression for $(x - 3)^2$?
What is the expression for $(x - 3)^2$?
Tap to reveal answer
$x^2 - 6x + 9$. Use $(a-b)^2 = a^2 - 2ab + b^2$.
$x^2 - 6x + 9$. Use $(a-b)^2 = a^2 - 2ab + b^2$.
← Didn't Know|Knew It →
Simplify $\frac{x^2 - 3x}{x - 3}$.
Simplify $\frac{x^2 - 3x}{x - 3}$.
Tap to reveal answer
$x$. Factor numerator: $\frac{x(x-3)}{x-3} = x$.
$x$. Factor numerator: $\frac{x(x-3)}{x-3} = x$.
← Didn't Know|Knew It →
What is the expression for the sum of $2x^2 + 3x$ and $x^2 - x$?
What is the expression for the sum of $2x^2 + 3x$ and $x^2 - x$?
Tap to reveal answer
$3x^2 + 2x$. Combine like terms: $(2+1)x^2 + (3-1)x = 3x^2 + 2x$.
$3x^2 + 2x$. Combine like terms: $(2+1)x^2 + (3-1)x = 3x^2 + 2x$.
← Didn't Know|Knew It →
Identify the leading coefficient in $5x^4 - 3x^2 + x$.
Identify the leading coefficient in $5x^4 - 3x^2 + x$.
Tap to reveal answer
- The coefficient of the highest degree term.
- The coefficient of the highest degree term.
← Didn't Know|Knew It →
Simplify the expression $\frac{x^2 - 9}{x + 3}$.
Simplify the expression $\frac{x^2 - 9}{x + 3}$.
Tap to reveal answer
$x - 3$. Factor numerator: $\frac{(x+3)(x-3)}{x+3} = x-3$.
$x - 3$. Factor numerator: $\frac{(x+3)(x-3)}{x+3} = x-3$.
← Didn't Know|Knew It →
What is the standard form of a quadratic polynomial?
What is the standard form of a quadratic polynomial?
Tap to reveal answer
$ax^2 + bx + c$. The general form where $a \neq 0$.
$ax^2 + bx + c$. The general form where $a \neq 0$.
← Didn't Know|Knew It →
State the expression for the difference of cubes $a^3 - b^3$.
State the expression for the difference of cubes $a^3 - b^3$.
Tap to reveal answer
$(a - b)(a^2 + ab + b^2)$. The factorization formula for $a^3 - b^3$.
$(a - b)(a^2 + ab + b^2)$. The factorization formula for $a^3 - b^3$.
← Didn't Know|Knew It →
Identify the expression equivalent to $\frac{x^3 - x}{x}$.
Identify the expression equivalent to $\frac{x^3 - x}{x}$.
Tap to reveal answer
$x^2 - 1$. Factor and cancel: $\frac{x(x^2-1)}{x} = x^2-1$.
$x^2 - 1$. Factor and cancel: $\frac{x(x^2-1)}{x} = x^2-1$.
← Didn't Know|Knew It →
What is the degree of the polynomial $4x^3 - 2x + 7$?
What is the degree of the polynomial $4x^3 - 2x + 7$?
Tap to reveal answer
$3$. The highest power of $x$ in the polynomial.
$3$. The highest power of $x$ in the polynomial.
← Didn't Know|Knew It →
Simplify the expression $\frac{x^2 - 9}{x + 3}$.
Simplify the expression $\frac{x^2 - 9}{x + 3}$.
Tap to reveal answer
$x - 3$. Factor numerator: $\frac{(x+3)(x-3)}{x+3} = x-3$.
$x - 3$. Factor numerator: $\frac{(x+3)(x-3)}{x+3} = x-3$.
← Didn't Know|Knew It →
Simplify the expression $\frac{x^2 - 4x}{x}$.
Simplify the expression $\frac{x^2 - 4x}{x}$.
Tap to reveal answer
$x - 4$. Factor out $x$: $\frac{x(x-4)}{x} = x-4$.
$x - 4$. Factor out $x$: $\frac{x(x-4)}{x} = x-4$.
← Didn't Know|Knew It →
What is the polynomial remainder theorem?
What is the polynomial remainder theorem?
Tap to reveal answer
Remainder of $f(x)$ divided by $x - a$ is $f(a)$. Substitute $x=a$ into $f(x)$ to find remainder.
Remainder of $f(x)$ divided by $x - a$ is $f(a)$. Substitute $x=a$ into $f(x)$ to find remainder.
← Didn't Know|Knew It →
State the remainder when $x^3 + 3x^2 - x + 5$ is divided by $x - 1$.
State the remainder when $x^3 + 3x^2 - x + 5$ is divided by $x - 1$.
Tap to reveal answer
- By remainder theorem, substitute $x=1$: $1+3-1+5=8$.
- By remainder theorem, substitute $x=1$: $1+3-1+5=8$.
← Didn't Know|Knew It →
What is the greatest common factor of $6x^2 + 9x$?
What is the greatest common factor of $6x^2 + 9x$?
Tap to reveal answer
3x. Factor out the GCF: $3x(2x + 3)$.
3x. Factor out the GCF: $3x(2x + 3)$.
← Didn't Know|Knew It →
Identify the expression equivalent to $\frac{x^2 + 2x}{x}$.
Identify the expression equivalent to $\frac{x^2 + 2x}{x}$.
Tap to reveal answer
$x + 2$. Factor out $x$: $\frac{x(x+2)}{x} = x+2$.
$x + 2$. Factor out $x$: $\frac{x(x+2)}{x} = x+2$.
← Didn't Know|Knew It →
What is the quotient of $\frac{x^2 - 4}{x - 2}$?
What is the quotient of $\frac{x^2 - 4}{x - 2}$?
Tap to reveal answer
$x + 2$. Factor numerator: $\frac{(x+2)(x-2)}{x-2} = x+2$.
$x + 2$. Factor numerator: $\frac{(x+2)(x-2)}{x-2} = x+2$.
← Didn't Know|Knew It →
What is the expression for the difference between $x^2 + 4x$ and $2x^2 + x$?
What is the expression for the difference between $x^2 + 4x$ and $2x^2 + x$?
Tap to reveal answer
$-x^2 + 3x$. $(x^2 + 4x) - (2x^2 + x) = -x^2 + 3x$.
$-x^2 + 3x$. $(x^2 + 4x) - (2x^2 + x) = -x^2 + 3x$.
← Didn't Know|Knew It →
Find the product of $(x + 1)$ and $(x - 1)$.
Find the product of $(x + 1)$ and $(x - 1)$.
Tap to reveal answer
$x^2 - 1$. Difference of squares formula gives $x^2 - 1^2$.
$x^2 - 1$. Difference of squares formula gives $x^2 - 1^2$.
← Didn't Know|Knew It →
What is the expression for the sum of $2x^2 + 3x$ and $x^2 - x$?
What is the expression for the sum of $2x^2 + 3x$ and $x^2 - x$?
Tap to reveal answer
$3x^2 + 2x$. Combine like terms: $(2+1)x^2 + (3-1)x = 3x^2 + 2x$.
$3x^2 + 2x$. Combine like terms: $(2+1)x^2 + (3-1)x = 3x^2 + 2x$.
← Didn't Know|Knew It →
State the zeroes of the polynomial $x^2 - 5x + 6$.
State the zeroes of the polynomial $x^2 - 5x + 6$.
Tap to reveal answer
2, 3. Factor as $(x-2)(x-3)$ and set each factor to zero.
2, 3. Factor as $(x-2)(x-3)$ and set each factor to zero.
← Didn't Know|Knew It →
What is the factored form of $x^2 - 9$?
What is the factored form of $x^2 - 9$?
Tap to reveal answer
$(x - 3)(x + 3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.
$(x - 3)(x + 3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.
← Didn't Know|Knew It →
Convert $f(x) = (x - 2)(x + 3)$ to standard form.
Convert $f(x) = (x - 2)(x + 3)$ to standard form.
Tap to reveal answer
$x^2 + x - 6$. Use FOIL: $(x-2)(x+3) = x^2 + 3x - 2x - 6$.
$x^2 + x - 6$. Use FOIL: $(x-2)(x+3) = x^2 + 3x - 2x - 6$.
← Didn't Know|Knew It →
Identify the leading coefficient in $5x^4 - 3x^2 + x$.
Identify the leading coefficient in $5x^4 - 3x^2 + x$.
Tap to reveal answer
- The coefficient of the highest degree term.
- The coefficient of the highest degree term.
← Didn't Know|Knew It →
What is the degree of the polynomial $4x^3 - 2x + 7$?
What is the degree of the polynomial $4x^3 - 2x + 7$?
Tap to reveal answer
- The highest power of $x$ in the polynomial.
- The highest power of $x$ in the polynomial.
← Didn't Know|Knew It →
What does it mean for a polynomial to be monic?
What does it mean for a polynomial to be monic?
Tap to reveal answer
Leading coefficient is 1. The coefficient of the highest degree term equals 1.
Leading coefficient is 1. The coefficient of the highest degree term equals 1.
← Didn't Know|Knew It →
What is the expression for the cube of $(x + 2)$?
What is the expression for the cube of $(x + 2)$?
Tap to reveal answer
$x^3 + 6x^2 + 12x + 8$. Use $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.
$x^3 + 6x^2 + 12x + 8$. Use $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.
← Didn't Know|Knew It →
What is the common denominator of $\frac{1}{x}$ and $\frac{1}{x^2}$?
What is the common denominator of $\frac{1}{x}$ and $\frac{1}{x^2}$?
Tap to reveal answer
$x^2$. Use the highest power of $x$ in both denominators.
$x^2$. Use the highest power of $x$ in both denominators.
← Didn't Know|Knew It →
Convert $\frac{3x + 6}{x + 2}$ to simplest form.
Convert $\frac{3x + 6}{x + 2}$ to simplest form.
Tap to reveal answer
- Factor and cancel: $\frac{3(x+2)}{x+2} = 3$.
- Factor and cancel: $\frac{3(x+2)}{x+2} = 3$.
← Didn't Know|Knew It →