Simplify a rational expression

Slides:

Advertisements

Similar presentations

EXAMPLE 3 Standardized Test Practice SOLUTION 8x 3 y 2x y 2 7x4y37x4y3 4y4y 56x 7 y 4 8xy 3 = Multiply numerators and denominators. 8 7 x x 6 y 3 y 8 x.

Advertisements

EXAMPLE 4 Solve a multi-step problem ICE SCULPTURES

Multiply complex numbers

EXAMPLE 6 Solve a multi-step problem TERRARIUM

EXAMPLE 4 Solve a multi-step problem

EXAMPLE 2 Use the scale factor of similar solids Packaging

Polynomial Multiplication

EXAMPLE 5 Solve a multi-step problem BLOOD VESSELS

EXAMPLE 3 Simplify an expression by dividing out binomials Simplify x 2 – 3x – 10 x 2 + 6x + 8. State the excluded values. SOLUTION x 2 – 3x – 10 x 2 +

Solve an absolute value inequality

Solve an equation by combining like terms

Simplifying a Variable Expression

Solve a compound inequality with and

Division by Polynomials

EXAMPLE 1 Collecting Like Terms x + 2 = 3x x + 2 –x = 3x – x 2 = 2x 1 = x Original equation Subtract x from each side. Divide both sides by x2x.

EXAMPLE 2 Identifying Equivalent Fractions

Solve a radical equation

Solve an equation with an extraneous solution

Warm Up #3 Find the exact value. 2. –√ √49 ANSWER –12 7 ANSWER

Algebra I Chapter 10 Review

Solve an equation with an extraneous solution

( ) EXAMPLE 3 Standardized Test Practice SOLUTION 5 x = – 9 – 9

EXAMPLE 2 Rationalize denominators of fractions Simplify

3.6 Solving Quadratic Equations

Chapter 9: Rational Expressions Section 9-1: Multiplying and Dividing Rationals 1.A Rational Expression is a ratio of two polynomial expressions. (fraction)

Aim: How do we multiply and divide rational expressions?

EXAMPLE 2 Multiply rational expressions involving polynomials Find the product 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x Multiply numerators and denominators.

Warm-Up Exercises 1. Right rectangular prism, side lengths 8 in., 5 in., and 10 in. 2. Right cone, radius 3 m, height 4 m ANSWER 340 in. 2 ; 400 in. 3.

Warm Up A shape has 5 faces, and 5 vertices how many edges does the shape have? A sphere has a radius of 7.5, what is its surface area and volume? What.

3.8 Warm Up 1. (18x³ - 24x² + 12x) ∕ 6x 2. (5x² + 7x – 6) ∕ (x + 2) 3. (9x² + 5x – 6) ∕ (x + 1)

Warm Up 1. Evaluate 5 2 – 3 ANSWER – Evaluate ANSWER Simplify 6a6a –4 b 0. ANSWER 6 a 4 4. Simplify 8x 3 y –4 12x 2 y –3. ANSWER.

5.3 Solving Quadratic Equations by Finding Square Roots.

1. √49 2. –√144 Lesson 4.5, For use with pages

1. 2. Lesson 8.3, For use with pages x – . Graph y = ANSWER

Chapter 12 Final Exam Review. Section 12.4 “Simplify Rational Expressions” A RATIONAL EXPRESSION is an expression that can be written as a ratio (fraction)

5.6 Solving Quadratic Function By Finding Square Roots 12/14/2012.

EXAMPLE 6 Solve a multi-step problem A = 13, x 1 – 0.015x and T = x 1 – 0.016x The amount A (in millions of dollars) spent on all advertising.

EXAMPLE 1 Identify similar solids Tell whether the given right rectangular prism is similar to the right rectangular prism shown at the right. a. b.

5.4 Multiply & Divide Rational Expressions

EXAMPLE 4 Solve a multi-step problem Computers You are buying a new computer and find 10 models in a store advertisement. The prices are $890, $750, $650,

8 4 Multiply & Divide Rational Expressions

EXAMPLE 4 Solve equations using nth roots Solve the equation. a. 4x 5 = 128 Divide each side by 4. x5x5 32= Take fifth root of each side. x=  32 5 Simplify.

Warm-Up #1 11/30 1. Write down these 3 formulas:

Solve an inequality using multiplication EXAMPLE 2 < 7< 7 x –6 Write original inequality. Multiply each side by –6. Reverse inequality symbol. x > –42.

EXAMPLE 5 Simplify a rational model 46 – 2.2x C = 100 – 18x + 2.2x 2 where x is the number of years since Rewrite the model so that it has only whole.

Do Now Pass out calculators. Have your homework out ready to check.

(x+2)(x-2).  Objective: Be able to solve equations involving rational expressions.  Strategy: Multiply by the common denominator.  NOTE: BE SURE TO.

3.8 Simplifying Rational Expressions p Vocabulary Rational Expression: ratio of 2 polynomials Rational Expression: ratio of 2 polynomials Excluded.

EXAMPLE 2 Simplify expressions by dividing out monomials Simplify the rational expression, if possible. State the excluded values. a. r 2r SOLUTION Divide.

Make a Model A box company makes boxes to hold popcorn. Each box is made by cutting the square corners out of a rectangular sheet of cardboard. The rectangle.

11.6 Adding and Subtracting Rational Expressions

EXAMPLE 2 Multiply by the LCD Solve. Check your solution. x – 2 x = SOLUTION x – 2 x = Multiply by LCD, 5(x – 2). 5(x – 2) x – 2 x 1 5.

Section 6.1 Rational Expressions Definition A rational expression is the ratio of two polynomials. Examples:

Warm-Up Exercises Perform the operation. 1. x x + 36 x 2 – x5x x 2 – 6x + 9 · x 2 + 4x – 21 x 2 + 7x ANSWERS x + 3 x – 12 ANSWERS 5 x – 3.

Factor the expression x – 5x2 3. x3 – 125 ANSWER 5x (2 – x)

Polynomial & Rational Inequalities

EXAMPLE 4 Solve a multi-step problem ICE SCULPTURES

EXAMPLE 2 Rationalize denominators of fractions Simplify

Completing the Square 8

EXAMPLE 1 Find excluded values

( ) EXAMPLE 3 Standardized Test Practice SOLUTION 5 x = – 9 – 9

Solve a quadratic equation

Find the surface area of a sphere

9.4: Rational Expressions

Find the surface area of the

Multiplying and Dividing Rational Numbers

Solve an inequality using subtraction

Warm Up #3 Find the exact value. 2. –√ √49 ANSWER –12 7 ANSWER

A rational expression is a quotient of two polynomials

Multiplying and Dividing Rational Numbers

Presentation transcript:

Simplify a rational expression EXAMPLE 1 Simplify a rational expression x2 – 2x – 15 x2 – 9 Simplify : SOLUTION x2 – 2x – 15 x2 – 9 (x +3)(x –5) (x +3)(x –3) = Factor numerator and denominator. (x +3)(x –5) (x +3)(x –3) = Divide out common factor. x – 5 x – 3 = Simplified form ANSWER x – 5 x – 3

EXAMPLE 2 Solve a multi-step problem A company makes a tin to hold flavored popcorn. The tin is a rectangular prism with a square base. The company is designing a new tin with the same base and twice the height of the old tin. Packaging • Find the surface area and volume of each tin. • Calculate the ratio of surface area to volume for each tin. • What do the ratios tell you about the efficiencies of the two tins?

Solve a multi-step problem EXAMPLE 2 Solve a multi-step problem SOLUTION Old tin New tin STEP 1 S = 2s2 + 4sh S = 2s2 + 4s(2h) Find surface area, S. = 2s2 + 8sh V = s2h V = s2(2h) Find volume, V. = 2s2h STEP 2 S V 2s2 + 4sh = s2h S V 2s2 + 8sh = 2s2h Write ratio of S to V. 2s(s + 4h) = 2s(sh) s(22 + 4h) = s(sh) Divide out common factor. 2s + 4h = sh s + 4h = sh Simplified form

EXAMPLE 2 Solve a multi-step problem STEP 3 2s + 4h sh > s + 4h Because the left side of the inequality has a greater numerator than the right side and both have the same (positive) denominator. The ratio of surface area to volume is greater for the old tin than for the new tin. So, the old tin is less efficient than the new tin.

Simplify the expression, if possible. GUIDED PRACTICE for Examples 1 and 2 Simplify the expression, if possible. 2(x + 1) (x + 1)(x + 3) 1. SOLUTION (x +1)(x + 3) 2(x +1) = 2(x + 1) (x + 1)(x + 3) Divide out common factor. = x + 3 2 Simplified form ANSWER x + 3 2

GUIDED PRACTICE for Examples 1 and 2 40x + 20 2. 10x + 30 SOLUTION = 20(2x +1) 10(x + 3) Factor numerator and denominator. 20(2x +1) 10(x + 3) = Divide out common factor. = 2(2x +1) x + 3 Simplified form 2(2x +1) x + 3 ANSWER

GUIDED PRACTICE for Examples 1 and 2 3. 4 x(x + 2) SOLUTION 4 x(x + 2) Simplified form x(x + 2) 4 ANSWER

GUIDED PRACTICE for Examples 1 and 2 x + 4 4. x 2 –16 SOLUTION x + 4 = Factor numerator and denominator. (x + 4) (x + 4)(x – 4) = Divide out common factor. 1 x – 4 = Simplified form ANSWER 1 x – 4

GUIDED PRACTICE for Examples 1 and 2 5. x2 – 2x – 3 x2 – x – 6 SOLUTION (x – 3)(x + 1) (x – 3)(x + 2) x2 – 2x – 3 x2 – x – 6 = Factor numerator and denominator. = (x – 3)(x + 1) (x – 3)(x + 2) Divide out common factor. x + 1 x + 2 = Simplified form x + 1 x + 2 ANSWER

GUIDED PRACTICE for Examples 1 and 2 2x2 + 10x 3x2 + 16x + 5 6. SOLUTION 2x2 + 10x 3x2 + 16x + 5 (3x + 1)(x + 5) 2x(x + 5) = Factor numerator and denominator. (3x + 1)(x + 5) 2x(x + 5) = Divide out common factor. 2x 3x + 1 = Simplified form ANSWER 2x 3x + 1

GUIDED PRACTICE for Examples 1 and 2 7. What If? In Example 2, suppose the new popcorn tin is the same height as the old tin but has a base with sides twice as long. What is the ratio of surface area to volume for this tin? SOLUTION Old tin New tin STEP 1 S = 2s2 + 4sh S = 2 (2s)2 + 4(2s)h Find surface area, S. = 8s2 + 8sh V = s2h V = (2s)2h Find volume, V. = 4s2h

GUIDED PRACTICE for Examples 1 and 2 STEP 2 S V 2s2 + 4sh = s2h S V Write ratio of S to V. 4s(2s + 2h) = 4s(sh) s(2s + 4h) = s(sh) Divide out common factor. = 2s + 4h sh 2s + 4h = sh Simplified form 2s + 4h = sh ANSWER