Lesson 4 Add Subtract multiply and divide rational

Lesson 4 Add, Subtract, multiply, and divide rational expressions

Warm up Find the product. • (x+6)(x-6) • (7 x-8)(7 x+8) 1. (x+6)²

Warm up • Divide 5 x³ + 20 x² - 10 x by 5 x

What is a rational expression? • A rational expression is an expression that can be written as a ratio of two polynomials. Ex. 8 -2 x x x 2 – 3 x x+5 x² + 2 x - 15

Excluded values • A rational expression is undefined when the denominator is 0. A number that makes the denominator 0 is an excluded value. • Let’s look at the examples before…are there any values that should be excluded? 8 -2 x x x 2 – 3 x x+2 x² + 2 x - 15

Examples • Find the excluded values, if any, of the expression. 1. 2. 9 x 5 x - 15 2. x -1 x² - 16 3. x + 6 x² + 4 x - 12 To find the excluded values, you have to factor the denominator (if possible). Then set each factor equal to ZERO and solve.

Simplest Form • A rational expression is in simplest form if the numerator and denominator have no factors in common other than 1. • Examples: Are these expressions in simplest form? 8 -2 x x 3 x - 9 x+5 x² + 2 x - 15

What about these? • Simplify, if possible. State the excluded values. 2 m 11 8 m(m-1) y+6 7 q² - 14 q 14 q²

Try these… • Simplify and state the excluded values. x² + 4 x – 21 x² + 13 x + 42 x² - 5 x + 6 x² -2 x -63

Guided Practice • Pp. 437 -438 1 -3 (all parts) • Rational Expressions Handout

Multiplying Rational Expressions • • • ½*¾= 2/5 * 5/4 = 2/3 * 5/4 = 2 x² * 1/x = 4 xy/5 * 20 y/4 x²

Multiply and Divide Rational Expressions with variables • Factor the numerators and denominators. • Simplify where possible (…Cancel) • Multiply the numerators and denominators straight across. • List restrictions (excluded values) if any… Example: x² + x – 6 10 x² - 20 x 5 x² + 15 x * x² - 2 x - 15

Examples 4 x² 2 x³ + 10 x – 48 x x² - 1 * 2 x² - 3 x + 1 * 4 x – 2 3 x + 18 (x+8)

Guided Practice • Textbook p. 441 -442

Let’s Divide • ¾ divided by ½ • 2/5 divided by 2 • 1/3 divided by 4/5

$So…to divide rational expressions… 1. Find the reciprocal of the second fraction. 2. Simplify$

So…to divide rational expressions… 1. Find the reciprocal of the second fraction. 2. Simplify according to your rules… 3. Multiply straight across…

Try these… 24 ÷ 6 5 x² 25 x² 7 x +21 ÷ 21 x + 63 30 x+2 3 x – 3 20 x² + 11 x + 18 x-1

Graphic Organizer • Fill in your graphic organizer “How do you multiply or divide rational expressions? ” • Work each example in the space provided.

Practice • Complete textbook pp. 441 -444 • Dividing rational expressions handout. • Multiplying rational exp handout.

Finding the LCM • Find the LCM of 6 x and 8 x² Step 1: Write the factors of each expression. 6 x = 2 * 3 * x 8 x² = 2 * 2 * x

Next… • Step 2: 6 x = 2 * 3 * x 8 x² = 2 *2 * x Circle the factors that both expressions have in common… • Step 3: List the common factors once…and then list every other factor… • Step 4: MULTIPLY 2 * x * 3 * 2 * x =24 x²

Guided Practice • Least Common Multiple Practice Handout • Complete 1 -3

Find the LCM of… • x² + 2 x – 8 and x² + 7 x + 12 Step 1: List the factors of each polynomial x² + 2 x – 8 = (x-2)(x+4) x² + 7 x + 12 = (x+3)(x+4) Step 2: Circle the common factors. Step 3: List the common factor once and then the other factors…then multiply… (x+4)(x-2)(x+3) There’s no need to multiply this out…Leave it as it is…

Guided Practice • Complete the remaining problems on the practice handout.

Add/Subtract Rationals • x+3 + x– 2 7 x 7 x • 5 x + 7 - 2 x – 9 3 x – 4 So, on these I need to add/subtract the numerators and just bring over the denominator, then simplify, if possible?

What if the denominators are not the same? • 11 + 15 12 x² 16 x 5 1. Find the LCD of the denominators…aka LCM. 2. Multiply top and bottom by missing factors… 3. Now that the denominators are the same…follow your steps for adding fractions…