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POLYNOMIALS and RATIONAL EXPRESSIONS Simplifying Rational Expressions Overview of Objectives, students should be able to: 1. Factor using sum and difference of cubes pattern. 2. Simply rational expressions via factoring. A C 3. Multiply and simplify rational expressions: ⋅ B D 4. Divide and simplify two rational expressions: A C ÷ B D 5. Add and subtract two rational expressions with common denominators. Main Overarching Questions: 1. How do you know you need to simplify an expression? 2. What do you do to simplify expressions? 3. For the sum and difference of cubes, how do you remember the formulas? Note: The rational operation and simplification objectives are similar to those in MAT 0024, however problems are usually more involved and rely more on factoring. If students are familiar with the basic operations a general discussion about how the process works might be more appropriate than reintroducing the material from scratch. Objectives: Activities and Questions to ask students: • • • • • • • Factor using sum and difference of cubes pattern. • • • • • Provide students with an example of the difference of SQUARES first: How did we factor 4x2 – 9? How did we factor 16x4 – 49y2? Recall when we multiplied: (3x+y)(3x-y) We recognized the pattern as (a + b)(a - b) = a2 - b2 How could you use this information to factor 4x2 – 9? (2x-3)(2x+3) Is it factored completely? Now, let’s look at the sum and difference of CUBES: o a3 – b3 = (a - b)(a2 + ab + b2) o a3 + b3 = (a + b)(a2 - ab + b2) What differences do you notice between the two formulas? How can you prove the formulas are true? Talk through until they agree that multiplying the right side out will give them the sum/difference of cubes. How would you recognize the sum or difference of cubes? Ask students to provide an example of what the sum of cubes would look like. Ask how do they know it is the sum of cubes. Ask students to provide an example of what the difference of cubes would look like. The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • • Give students a worksheet to work in groups. This worksheet should include all the possible factoring problems. Discuss answers. Have students give answers and defend their answers. Always ask is it factored completely? Note: This is a review from MAT 0024. However, the students should be better skilled at factoring. Simply rational expressions via factoring. • • • Multiply and simplify rational expressions: A C ⋅ B D 16 ? What process or operations did you use in your reduction? 30 2x 3 2 ⋅ x ⋅ x ⋅ x 2 What about 4 = = ? What process or operations did you use in your reduction? x⋅x⋅x⋅x x x How do you reduce the fraction • How do you know when it is reduced to lowest terms? • Give another example: how could you simplify • example. What processes do you need in order to simplify a rational expression? 2x 2 + 4x ? Remember how you simplified the last 6 x 2 + 12 x • x2 − x − 2 How could you simplify 2 ? x + 5x + 4 • What happens to the numerator when simplifying: • • Do you see a pattern in the last two examples? (i.e. when the numerator “disappears” we must write a 1). Give a few simplification problems requiring factoring differences of squares, product/sum, and quadratic trinomials with leading coefficients not equal to 1. If students have trouble simplifying, review the process above. • How do you multiply fractions? i.e. • Give students several numeric examples. Do you see a pattern or process to multiply fractions? • • • x5 x−5 ? What about 2 ? 8 x x − 7 x + 10 4 7 ⋅ 5 8 P R P⋅R . Then, simplify. ⋅ = Q S Q⋅S 2 x−5 Have students work the example: ⋅ x−6 3 4 7 4/ 7 1 7 7 Was there a shortcut to our first example? ⋅ = ⋅ = ⋅ = 5 8 5 8/ 5 2 10 Have students draw out the process that The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • • • Divide and simplify two rational expressions: A C ÷ B D • Add and subtract two rational expressions with common denominators. • • x−6 4 ? ⋅ 3 x−6 6y + 2 1− y Have students work the example: ⋅ y 2 −1 3y 2 + y 3 7 How do you divide fractions? ? i.e. ÷ 4 8 How can we use the shortcut to multiply: Give students several numeric examples. Do you see a pattern or process to divide fractions? P R P S P⋅S . Then, simplify. ÷ = ⋅ = Q S Q R Q⋅R • Have students draw out the process that • Give students an example to use this process like: • If you want to use the shortcut from multiplication, at what step would you perform it? • How do you add fractions with the same denominator? i.e. • Give students several numeric examples. Do you see a pattern or process to add fractions? • How do you subtract fractions with the same denominator? i.e. • Give students several numeric examples. Do you see a pattern or process to subtract fractions? • • • y 2 + 4 y − 21 y 2 + 14 y + 48 ÷ y 2 + 3 y − 28 y 2 + 4 y − 32 3 1 + 8 8 5 1 − 8 8 P R P±R . Then, simplify. ± = Q Q Q 3y 6 Have students work out the example: 2 − 2 y + 3 y − 10 y + 3 y − 10 Have students draw out the process that x 2 + 6x + 2 2x − 1 Have students work out the example: − 2 2 x + x−6 x + x−6 The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. Adding and Subtracting Rational Expressions with Unlike Denominators Overview of Objectives, students should be able to: Main Overarching Questions: 1. How do you find the LCD of a list of rational expressions? 1. Find the least common denominator given two or more rational expressions 2. Add and subtract two or more rational expressions with different denominators 2. How do you add or subtract rational expressions that contain different denominators? 3. Why do you have to get common denominators before adding/subtracting rational expressions? 4. How is adding/subtracting rational expressions similar to adding/subtracting fractions? Objectives: • Find the least common denominator given two or more rational expressions Activities and Questions to ask students: 2 1 , . 3 6 • How do you find the least common denominator? • Give several numeric examples. Do you see a pattern or process? • What is the least common denominator of 2 3 , ? Write down the process you used to find x2 y the least common denominator. • Add and subtract two or more rational expressions with different denominators 2 4x . If students try to piece together terms to get , 2 x − 5 x − 25 • What about finding the LCD of • the LCD, remind them to think about the process they used to find the LCD with numbers (i.e. factoring) Give several more examples involving binomial and trinomial denominators. • When is this process useful? For what purpose could we use this? • How do you write equivalent fractions? Give a numeric example: • • Give several numeric examples. Do you see a pattern or process? Ask students what a cross product is. Can we use this to solve the problem another way? Do we get the same answer? 3 ? = . 4 8 The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. 5 ? = 3 y 12 y 2 • Have students work the example: • Does using cross products work in this case? When can we use cross products? When could you use this process to add or subtract fractions that do not have the same denominator? • • • 3x ? = 2 x + 5 x − 25 2 1 How do you add fractions with different denominators? + 3 6 Have students work another example, rewrite • • Give several numeric examples. Do you see a pattern or process? How can you use what you have already learned to add or subtract fractions with different denominators? • Have students work the example: problem. 3 4 . Write down the process you used to solve the + 2 4x 2x 4 3 2y + 9 2 , followed by: 2 + − x − 2 x −1 y − 7 y + 12 y − 3 • Give a harder example to work: • Have students work in groups on a worksheet of problems containing a mixture of addition, subtract, multiplication, and division problems. Is there more than one way to solve each problem? Explain. • Complex Fractions and Polynomial Division Overview of Objectives, students should be able to: 1. Simplify complex rational expressions 2. Divide two polynomials a. Divide two polynomials by simplifying the resulting rational expression b. Divide two polynomials by using long division Objectives: Main Overarching Questions: 1. How do you simplify complex fractions? 2. How do you divide polynomials? Activities and Questions to ask students: The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • Simplify complex rational expressions • • fractions they see in the complex fraction. Ask students what operation is represented between the two fractions (i.e. the fraction bar represents division) • Then ask students to think about we divided fractions. • Have students summarize the process. • 1 1 + x y ? What if we have a more complicated complex fraction like 1 1 − x y • • • • • • Divide two polynomials o Divide two polynomials by simplifying the resulting rational expression o Divide two polynomials by using long division x Present students with a simpler complex fraction: 2 . Ask students to write down what 4x 5 • P R P S P⋅S ÷ = ⋅ = Q S Q R Q⋅R Ask students to describe the differences between this complex fraction and the first one. How many fractions do I have in the numerator? How many in the denominator? How many did I have before? Once students realize there is now more than one fraction in the numerator and denominator, ask them how they might combine the two fractions in the numerator into one fraction. Ask students to recall what concepts they have studied that allows them to combine fractions with addition and subtraction. Ask students to summarize the process (i. e. combine all fractions in the numerator/denominator using LCD, write as division of two fractions, convert to multiplication and simplify). Give a group worksheet of one or two simpler complex fractions and one or two of the more complicated fractions. 4 5x2 . Hopefully they will notice the + 5x 5x 4 + 5x2 common denominators and simply add the numerators: . 5x Ask students to add the two fractions together: The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • Next ask them how they might simplify 3 + 4x . Is there a way to break up the fractions? If 8x students have trouble, ask them to consider how we combined the two fractions together from before. How do we reverse the process? A+ B A B = + C C C • Have students draw the conclusion that • • Give students a few practice problems to try. Ask students to next consider what happens if we have more than one term in the x 2 − 7 x + 12 . Can we split up the denominator? If students say yes, ask x −5 2 3 them to combine together + , using a common denominator. Ask them once again to x 5 denominator like • • consider if they can split up the denominator. Once students are convinced that they cannot split the denominator, suggest using long division to divide. Give students all the terminology before beginning. As a good analogy, complete the steps to long division concurrently with an example from arithmetic. Have students compare/contrast each step of polynomial long division with the arithmetic version. Give students a few long division problems to try on their own. Solve Rational Equations Overview of Objectives, students should be able to: 1. Solve Rational Equations Objectives: • Solve rational equations. Main Overarching Questions: 1. How do you solve rational equations? 2. What is an extraneous solution? Activities and Questions to ask students: 1 3 +2=. x x • Give students a simple example of a rational equation: • Ask students to observe what value of x is “not allowed.” If they are not sure, says we are never allowed to have ‘blank’ in the denominator.” The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government. • • Next ask them how they might proceed to solve the equation. What is different about this equation when compared with other equations we have solved? (We have fractions) How can we clear our fractions? What one quantity can we multiply by to ensure all fractions are cleared? 1 3 +2=. x x • Ask students what the LCD is in the previous example • • • Do we have to multiply the 2 by the LCD? Why? Have students multiply through by LCD to clear the fraction. What kind of equation is left over? How do we solve it? Have students summarize the process of solving radical equations. • Give students another rational equation: • What solution did you get? Does this solution work when plugged in? How could we have known the solution would not work? Define the solution x = 3 in the previous example as an extraneous solution. Remind students to always list out restrictions on the variable at the beginning of the problem. • x 3 = + 9 . Ask them to solve it. x −3 x −3 The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.