Chapter 6 Rational Expression and Equations How are rational expressions simplified and rational equations solved?

ACTIVATION Review Yesterday’s Warm-up

6-1 Multiplying and Simplifying EQ: How do you multiply and simplify rational expressions?

Simplify the following Is the following a valid fraction? Why or Why not?

Simplest form—when all common factors have been removed Example y = x x + 25 x 2 + 9x + 20 If the denominator cannot equal zero what do we do with when we have a variable?

Multiplying Expressions Simplify: 2x 2 + 7x + 3 x x - 4 x 2 + 8x + 15

Dividing Expressions Invert the fraction behind the division sign and multiply 6x + 6y 18 x – y 5x – 5y ÷

Homework PAGE(S): 248 NUMBERS: 2 – 32 evens

ACTIVATION Review Yesterday’s Warm-up

6-2 Addition and Subtraction EQ: How do you add and subtract rational expressions?

Activation Add Add x+1 x -1 What is required to add and subtract fractions?

Examples

Homework PAGE(S): NUMBERS: 2 – 30 evens

ACTIVATION Review Yesterday’s Warm-up

6-3 Complex Rational Expressions EQ: How do you simplify complex rational expressions?

How do you simplify the following:

Evaluating Complex Expressions Is nothing more than dividing fractions means

Homework PAGE(S): 258 NUMBERS: 6 – 20 even

ACTIVATION Review Yesterday’s Warm-up

6-4 Division of Polynomials EQ: How do you divide polynomials?

What procedures would you use to solve the following problem:

Can we translate this to algebraic equations

Example 2

Homework PAGE(S): 262 NUMBERS: 12 – 22 even

ACTIVATION Review Yesterday’s Warm-up

6-5 Synthetic Division EQ: What is synthetic division?

Long division can be cumbersome Patterns were seen that can be used when the divisor is linear (x 3 +3x 2 – x – 3) ÷ (x – 1)

Long division can be cumbersome Patterns were seen that can be used when the divisor is linear (x 3 +3x 2 – x – 3) ÷ (x – 1)

The remainder theorem helps to determine roots as well but does not give the remaining factors/roots Example: Given: f(x)= x 3 + 4x 2 + 4x are 2, -1 or 0 roots?

Homework PAGE(S): 265 NUMBERS: even

ACTIVATION Review Yesterday’s Warm-up

6-6 Solving Rational Equations EQ: How do you solve a rational equation?

Solve 5 = 15. 2x -2 x 2 – 1 Check for any values that cause the fraction to be undefined Solve x = x + 6. x - 1 x + 3 Check for any values that cause the fraction to be undefined

Example Remember to check for extraneous values

Example Remember to check for extraneous values

Homework PAGE(S): 269 NUMBERS: 4, 8, 12, 14, 24, 26

ACTIVATION Review Yesterday’s Warm-up

6-7 Using Rational Equations EQ: How do you translate word problems into rational equations that can be solved?

Examples: Antonio, an experienced shipping clerk, can fill a certain order in 5 hours. Brian a new clerk, needs 9 hours to do the same job. Working together, how long would it take them to fill the order? Work problems use inverses: Antonio: 5 hrs Brian: 9 hrs Total job: t hrs

Example The speed of the stream is 4 km/hr. A boat travels 6 km upstream in the same time it takes to travel 12 km downstream. What is the speed of the boat in still water? DistanceRateTime Upstream downstream 6 kmX – rT 12X + rT

Homework PAGE(S): NUMBERS: 2, 4, 6, 14, 20

6-8 Formulas EQ: How do you solve rational formulas for a specified variable?

Example

Homework PAGE(S): 278 NUMBERS: 4, 8, 12, 16

6-9 Variation and Problem Solving EQ: What are direct and inverse variation?

Vocabulary Direct variation—when the ratio of two numbers is constant y = kx n Inverse variation—when the product of a series of numbers is constant y = Joint variation—multiple direct variations k—the constant of variation

Y varies directly with the square of x. What is the value of y when x = 3, if x=2 when y = 12. y = kx n

Example Y varies inversely with the square of x. What is the value of y when x = 3. If x=2 when y = 9 y = k x n

Example Example: Set up the following problem. Y varies inversely as x but directly as the cube of v. What is the value of y when x = 2 and v = 3, if y = 16 when x=3 and v = 2

Joint Variation when y varies jointly with x and the square of z. Find the general equation if y= 12 when x = 2 and z = 3.

Homework PAGE(S): 283 NUMBERS: 4 – 24 by 4’s

Homework PAGE(S): 289 NUMBERS: all