1. Fundamental Concepts of Algebra

Divide Rational Expressions

Rational Expressions

Rational Expressions:  Multiplying and Dividing.  Ex 1

Simplifying Complex Fractions

Find the domain of each rational expression. 
(2x - 4) / (x + 7)

Simplifying Complex Fractions - Ex 3

Adding and Subtracting Rational Expressions

Simplify each complex fraction. [1 + (1/x)] / (1-1/x)

Rational Expressions: Writing in Lowest Terms - Ex 2

Simplify each complex rational expression. (1/x-1/2)/(1/3-x/6)

Simplifying Complex Fractions - Ex 1

Simplify each complex fraction. [ 2 + 2/(1+x) ] / [ 2 - 2/(1-x) ]

Rational Expressions:  Adding and Subtracting.  Ex 1

❖ Finding all the Zeros of a Polynomial - Example 3 ❖

Simplify each complex fraction. [ 1/(x^3-y^3) ] / [ 1/(x^2 -y^2) ]

Simplify each rational expression. Also, list all numbers that must be excluded from the domain. [x^3+2x^2]/[x+2]

Evaluate each expression for x = -4 and y = 2. |-3x + 4y|

Perform the indicated operation, and write each answer in lowest terms. 2x/5 * 10/x^2

In Exercises 111–120, use the order of operations to simplify each expression. 10^2−100÷5^2⋅2−3

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). 4+5(x-7)^3, for x =9

Simplify each complex rational expression. [3-1/(x+3)]/[3+1/(x+3)]

Simplify each complex fraction. (2 - 2/y) / (2 + 2/y)

Simplify each complex fraction. [ (y+3)/y - 4/(y-1) ] / [ y/(y - 1) + 1/y ]

Simplify each rational expression. Assume all variable expressions represent positive
real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [2(2x-3)^1/3 - (x-1)(2x-3)^-2/3] / [(2x-2)^-2/3]

Write each rational expression in lowest terms. x^3 + 64 / x + 4

Write each rational expression in lowest terms. 3(3 - t) / (t + 5)(t - 3)

Find the domain of each rational expression. (3x + 7) / (4x + 2)(x - 12)

Evaluate each expression for x = -4 and y = 2. 4|x| + 4|y| / |x|

Rewrite each expression using the distributive property and simplify, if possible. 15x-10x

Multiply or divide, as indicated. (y^3 + y^2)/7 * 49/(y^4 + y^3)

Add or subtract, as indicated. (7x + 8)/(3x + 2) - (x + 4)/(3x + 2)

In Exercises 97–102, write each algebraic expression without parentheses. 1/3(3x)+[(4y)+(−4y)]

Perform the indicated operation, and write each answer in lowest terms. 
2x/5 + x/4

Finding the Domain of an Expression Involving Fractions - Example 3

How to Find Restrictions on Rational Expressions

Algebra - Complex Fractions

Multiply or divide, as indicated. (x^2 - y^2)/(x - y)^2 * (x^2 - xy + y^2)/(x^2 - 2xy + y^2) ÷ (x^3 + y^3)/(x - y)^4

Multiply or divide as indicated. [(x^2+6x+9)(x+3)]/[(x^2-4)(x-2)]

In Exercises 111–120, use the order of operations to simplify each expression. 6(−4)−5(−3)/(9−10)

Subtracting rational expressions

Simplify each expression. -1/4(20m+8y-32ɀ)

Multiply or divide, as indicated. (2k + 8)/6 ÷ (3k + 12)/2

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x^2-6x+3, for x=7

In Exercises 85–96, simplify each algebraic expression. 5(3y−2)−(7y+2)

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). 6x-y, for x=3 and y=8

Rational Expressions:  Multiplying and Dividing.  Ex 3

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x^2+3x, for x=8

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x^2-3(x-y), for x=8 and y=2

In Exercises 85–96, simplify each algebraic expression. 18x^2+4−[6(x^2−2)+5]

Simplify each rational expression. Assume all variable expressions represent positive
real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [(x^2 +1)^4(2x) - x^2(4)(x^2+1)^3(2x)] / [(x^2+1)^8]

Find the domain of each rational expression. (x + 3) / (x - 6)

Dividing rational expressions

Rational Expressions:  Adding and Subtracting.  Ex 2

Add or subtract, as indicated. 3/(a - 2) - 1/(2 - a)

In Exercises 33–68, add or subtract as indicated. 4/(x^2+6x+9) + 4/(x+3)

In Exercises 69–82, simplify each complex rational expression. (1/(x+h)^2 − 1/x^2)/h

Find the domain of each rational expression. 12/ (x2 + 5x + 6)

Rational Expressions:  Multiplying and Dividing.  Ex 2

Multiply  Rational Expressions

Simplify each rational expression. Assume all variable expressions represent positive
real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [4(x^2- 1)^3 + 8x(x^2-1)^4] / [16(x^2-1)^3]

In Exercises 33–68, add or subtract as indicated. 3/(2x+4) + 2/(3x+6)

Rational Expressions and Domain

Find the domain of each rational expression. 3 / x2 - 5x - 6

Add or subtract, as indicated. 1/6m + 2/5m + 4/m

Simplify each complex fraction. [ 6/(x^2-25) + x ] / [ 1/(x - 5) ]

Simplify each complex fraction. [ 1/(a^3+b^3) ] / [ 1/(a^2 + 2ab + b^2) ]

Add or subtract as indicated. [(2x-7)/(x^2-9)]-[(x-10)/(x^2-9)]

Perform the indicated operation, and write each answer in lowest terms. y^3/8 ÷ y/4

Write each rational expression in lowest terms. 8k + 16 / 9k + 18

In Exercises 85–96, simplify each algebraic expression. 5(3x+4)−4

Simplify each complex fraction. [ 1/(y+3) - 1/y ] / (1/y)

Rewrite simple rational expressions using inspection

Perform the indicated operation, and write each answer in lowest terms. 4/x-y - 9/x-y

Simplify each complex fraction. [ 1/(x+1) - 1/x ] / (1/x)

Evaluate each expression for x = -4 and y = 2. 2|y| - 3|x| / |xy|

Add or subtract as indicated. 12/25x^2+11/15x^3

Evaluate each expression for x = -4 and y = 2. |-8y + x| / -|x|

Write each rational expression in lowest terms. 8m^2 + 6m - 9 / 16m^2 - 9

Multiply or divide, as indicated. (xz - xw + 2yz - 2yw)/(z^2 - w^2) * (4z + 4w + xz + wx)/(16 - x^2)

In Exercises 33–68, add or subtract as indicated. x/(x^2−2x−24) − x/(x^2−7x+6)

❖ Finding the Domain of a Function Algebraically (No graph!) ❖

Evaluate each expression for x = -4 and y = 2. |-5y + x|

Evaluate each expression for x = -4 and y = 2. |3x - 2y|

Perform all indicated operations, and write each answer with positive integer exponents. [ (x^-2 + y^-2)/ (x^-2 - y^-2) ] * [ (x+y)/(x-y) ]

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). 7+5x, for x = 10

Find the domain of each rational expression. x^3 - 1 / x - 1

Finding the Domain of an Expression Involving Fractions - Example 2

Use specific values for x and y to show that, in general, 1/x + 1/y is not equivalent to 1 / x + y.

Perform all indicated operations, and write each answer with positive integer exponents. {x- 9y^-1}/{ (x-3y^-1)(x+3y^-1)}

Simplifying Complex Fractions - Ex 2

In Exercises 85–96, simplify each algebraic expression. 2(5x−1)+14x

Manipulating Rational Expressions: Simplification and Operations

Add or subtract, as indicated. 3/2k + 5/3k

Add or subtract, as indicated. (17y + 3)/(9y + 7) - (-10y - 18)/(9y + 7)

In Exercises 69–82, simplify each complex rational expression. (x/3−1)/(x−3)

Simplify each complex fraction. [ (x+4)/x - 3/(x-2) ] / [ x/(x - 2) + 1/x ]

In Exercises 111–120, use the order of operations to simplify each expression. 8−3[−2(2−5)−4(8−6)]

Multiply rational expressions

Write each rational expression in lowest terms. m^2 - 4m + 4 / m^2 + m - 6

Simplify each complex fraction. [ m - 1/(m^2-4) ] / [ 1/(m + 2) ]

Multiply or divide as indicated. [(x^2-5x-24)/(x^2-x-12)]/[(x^2-10x+16)/(x^2+x-6)]

Rational Expressions: Writing in Lowest Terms - Ex 1

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). (2x+3y)/(x+1), for x=-2 and y=4

Simplify each complex fraction. [ y + 1/(y^2-9) ] / [ 1/(y + 3) ]

Simplify each complex fraction. [ (-2)/(x+h) - (-2)/x ] / h

Find the domain of each rational expression. 
x^2 - 1 / x + 1

Rewrite polynomial expressions by factoring

Multiply or divide, as indicated. ac + ad + bc + bd/a^2 - b^2 * a^3 - b^3/2a^2 + 2ab + 2b^2

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). 5(x+2)/(2x-14), for x=10

Simplify each complex fraction. [ 1 + 1/(1-b) ] / [ 1 - 1/(1+b) ]

Adding rational expressions

Add or subtract, as indicated. (x + y)/(2x - y) - 2x/(y - 2x)

Multiply or divide, as indicated. (m^2 + 3m + 2)/(m^2 + 5m + 4) ÷ (m^2 + 5m + 6)/(m^2 + 10m + 24)

Multiply or divide, as indicated. 15p^3/9p^2 * 12p/10p^3

Write each rational expression in lowest terms. 8x^2 + 16 / 4x^2

Add or subtract, as indicated. 5/12x^2y - 11/6xy

Simplify each complex fraction. [ 1/[(x+h)^2 + 9] - 1/(x^2+9)] / h

Simplify each complex fraction. [ 3/(p^2-16) + p ] / [ 1/(p - 4) ]

In Exercises 85–96, simplify each algebraic expression. 7−4[3−(4y−5)]

In Exercises 69–82, simplify each complex rational expression. (1/x + 1/y)/(x+y)

Finding all the Zeros of a Polynomial - Example 2

Finding the Domain of an Expression Involving Fractions - Example 1

Add or subtract, as indicated. 4/x + 1 + 1/x^2 - x + 1 - 12/x^3 + 1

In Exercises 69–82, simplify each complex rational expression. (3/(x−2) − 4/(x+2))/(7/x^2−4)

Add or subtract, as indicated. 1/a + b/a^2

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