Algebra 2 Simplifying Rational Expressions

Questions and Answers

What is the simplified form of $\frac{18}{2x-6}$?

$\frac{9}{x-3}$ (correct)

$\frac{9}{2(x-3)}$

$\frac{6}{x-3}$

$\frac{18}{2(x-3)}$

What is the simplified form of $\frac{x^2-4}{(x-2)(x+1)}$?

$\frac{x^2-2}{x+1}$

$\frac{x-2}{x+1}$

$\frac{x+4}{x+1}$

$\frac{x+2}{x+1}$ (correct)

What is the simplified form of $\frac{(3x^2-24x)}{(3x^2+12x)}$?

$\frac{x-8}{x+4}$ (correct)

$\frac{3(x-8)}{3(x+4)}$

$\frac{3x^2-24x}{3x^2+12x}$

$\frac{x-8}{3(x+4)}$

What is the simplified form of $\frac{3x^2}{x+2} \div \frac{3x}{x^2-4}$?

$\frac{x(x-2)}{x+2}$ Signup and view all the answers

What is the simplified form of $\frac{(x^2+2x)}{6x} \div \frac{(x^2-4)}{3x^2}$?

$\frac{x+2}{2(x-2)}$ Signup and view all the answers

What is the simplified form of $\frac{(x^2-5x-24)}{(x+1)} \times \frac{(x^2-6x-7)}{(x+3)}$?

$\frac{(x-8)(x+4)}{(x+1)(x+3)}$ Signup and view all the answers

What is the simplified form of $\frac{(x^2+19x+84)}{(4x-4)} \times \frac{(2x-2)}{(x^2+9x+14)}$?

$\frac{(x+12)}{4(x-1)}$ Signup and view all the answers

What is the simplified form of $\frac{(x^2-5x+4)}{(2x-8)} \div \frac{(3x^2-3x)}{(x^2)}$?

$\frac{x-1}{(2x-4)}$ Signup and view all the answers

What is the simplified form of $\frac{(10x^2+15x)}{(35x^2-5x)}$?

$\frac{2(2x+3)}{7x(7x-1)}$ Signup and view all the answers

What is the simplified form of $\frac{(x^4+x^3-2x^2)}{(x^4-x^3)}$?

$\frac{x^2(x+1)}{x^3(x-1)}$ Signup and view all the answers

What is the simplified form of $\frac{x^5}{(x-6)} \times \frac{(x^2-6x)}{(x^8)}$?

$\frac{1}{x^3(x-6)}$ Signup and view all the answers

What is the simplified form of $\frac{(x^2-5x-24)}{(6x+2x^2)} \times \frac{5x^2}{(8-x)}$?

$\frac{-5x(x+4)}{2(-4+x)}$ Signup and view all the answers

What is the simplified form of $\frac{(x-5)}{(10x-2)} \times \frac{(25x^2-1)}{(x^2-10x+25)}$?

$\frac{(x-5)(5x+1)}{(10(x-2))}$ Signup and view all the answers

What is the simplified form of $\frac{(3x+6)}{(x^2-9)} \div \frac{(6x^2+12x)}{(4x+12)}$?

$\frac{2(x+2)}{(x-3)(x+3)}$ Signup and view all the answers

What is the simplified form of $\frac{(9-x^2)}{(x^2+5x+6)} \div \frac{(2x-6)}{(5x+10)}$?

$\frac{-(x-3)}{2(x+2)}$ Signup and view all the answers

What is the simplified form of $\frac{(2x-7)}{3} + 4$?

$\frac{(2x+12)}{3}$ Signup and view all the answers

What is the simplified form of $\frac{2}{(x+2)} - \frac{3}{2x}$?

$\frac{4x-6}{2x(x+2)}$ Signup and view all the answers

What is the simplified form of $\frac{(3x+7)}{(x-3)(x+3)} - \frac{3}{(x-3)}$?

$\frac{6x-7}{(x^2-9)}$ Signup and view all the answers

What is the simplified form of $\frac{(5x^2+4x-16)}{(x-4)(x+1)}$?

$\frac{5x-4}{(x+1)}$ Signup and view all the answers

What is the simplified form of $\frac{(2x^2+5x-2)}{(x-5)(x+2)}$?

$\frac{(2x-1)}{(x-5)}$ Signup and view all the answers

What is the simplified form of $\frac{(x+2)}{(x-3)} + \frac{(2x+2)}{(x^2-2x-3)}$?

$\frac{(3x+2)}{(x-3)}$ Signup and view all the answers

What is the simplified form of $\frac{(x+12)}{(x+4)(x-3)(x+2)}$?

$\frac{3}{(x^2+x-12)} - \frac{2}{(x^2+6x+8)}$ Signup and view all the answers

What is the simplified form of $\frac{(13x-47)}{(x-3)(x-5)}$?

$\frac{4}{(x-3)} + \frac{9}{(x-5)}$ Signup and view all the answers

Study Notes

Simplifying Rational Expressions

Key skill in algebra is simplifying rational expressions to their simplest forms without altering their values.

Expressions and Simplifications

9/(x-3) simplifies to 18/(2x-6) by factoring the denominator.
(x+2)/(x+1 simplifies to (x²-4)/(x-2)(x+1) by recognizing the difference of squares.
(x-8)/(x+4) simplifies from (3x²-24x)/(3x²+12x) through common factor extraction.
x(x-2) is achieved as 3x²/(x+2) ÷ 3x/(x²-4) by multiplying the first fraction by the reciprocal of the second.
x²/2(x-2) results from simplifying (x²+2x)/(6x) ÷ (x²-4)/(3x²) through factor cancellation.
(x-8)(x-7) simplifies from (x²-5x-24)/(x+1) × (x²-6x-7)/(x+3) by resolving quadratic factors.

Additional Simplifications

(x+12)/2(x+2) arises from (x²+19x+84)/(4x-4) × (2x-2)/(x²+9x+14) through division and factorization.
1/6x represents the simplified form of (x²-5x+4)/(2x-8) ÷ (3x²-3x) after factoring.
(2x+3)/(7x-1) comes from simplifying (10x²+15x)/(35x²-5x) through common factors.
(x+2)/x is derived from (x⁴+x³-2x²)/(x⁴-x³) by factoring out x.

Further Examples

1/x² is derived from the simplification of x⁵/(x-6) × (x²-6x)/x⁸ through canceling x factors.
-5x/2 results from (x²-5x-24)/(6x+2x²) × 5x²/(8-x) by canceling common factors.
(5x+1)/2(x-5) simplifies from (x-5)/(10x-2) × (25x²-1)/(x²-10x+25) through factoring.

Final Cases

2/x(x-3) simplifies from (3x+6)/(x²-9) ÷ (6x²+12x)/(4x+12) by factoring both numerator and denominator.
-5/2 results from simplifying (9-x²)/(x²+5x+6) ÷ (2x-6)/(5x+10) via factor cancellation.
(2x+5)/3 simplifies from (2x-7)/3 + 4 by combining like fractions.
(x-6)/2x(x+2) comes from 2/(x+2) - 3/2x, requiring a common denominator.
(3x+7)/(x-3)(x+3) simplifies from 3/(x-3) - 2/(x²-9) using common denominators.
(5x²+4x-16)/(x-4)(x+1) results from 4x/(x-4) + (x+4)/(x+1) through a common denominator.
(x²+x+1)/(x+1)² simplifies from 1/(x²+2x+1) + x/(x+1) by recognizing perfect square factors.
(2x²+5x-2)/(x-5)(x+2) is generated from simplifying (2x+1)/(x-5) - 4/(x²-3x-10).
(x+2)/(x-3) results from the simplification of x/(x-3) + (2x+2)/(x²-2x-3) collecting similar fractions.
(x+12)/(x+4)(x-3)(x+2) simplifies from 3/(x²+x-12) - 2/(x²+6x+8) through factoring.
(13x-47)/(x-3)(x-5) results from simplifying 4/(x-3) + 9/(x-5) requiring a common denominator.

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Test your knowledge of simplifying rational expressions with this set of flashcards. Each card presents an expression and its simplified form for you to master the concepts of algebra. Perfect for reinforcing skills in Algebra 2.