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Five-Minute Check (over Lesson 8–1) CCSS Then/Now Example 1: LCM of Monomials and Polynomials Key Concept: Adding and Subtracting Rational Expressions Example 2: Monomial Denominators Example 3: Polynomial Denominators Example 4: Complex Fractions with Different LCDs Example 5: Complex Fractions with Same LCD Lesson Menu

A. –3rt B. –3r C. 3rt2 D. 4r 5-Minute Check 1

A. –3rt B. –3r C. 3rt2 D. 4r 5-Minute Check 1

A. B. C. D. 5-Minute Check 2

A. B. C. D. 5-Minute Check 2

A. B. C. D. 5-Minute Check 3

A. B. C. D. 5-Minute Check 3

A. B. C. D. 5-Minute Check 4

A. B. C. D. 5-Minute Check 4

A. B. C. D. 5-Minute Check 5

A. B. C. D. 5-Minute Check 5

A. B. C. D. 5-Minute Check 6

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Mathematical Practices Content Standards A.APR.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. CCSS

You added and subtracted polynomial expressions. Determine the LCM of polynomials. Add and subtract rational expressions. Then/Now

A. Find the LCM of 15a2bc3, 16b5c2, and 20a3c6. LCM of Monomials and Polynomials A. Find the LCM of 15a2bc3, 16b5c2, and 20a3c6. 15a2bc3 = 3 ● 5 ● a2 ● b ● c3 Factor the first monomial. 16b5c2 = 24 ● b5 ● c2 Factor the second monomial. 20a3c6 = 22 ● 5 ● a3 ● c6 Factor the third monomial. LCM = 3 ● 5 ● 24● a3 ● b5 ● c6 Use each factor the greatest number of times it appears. Example 1A

= 240a3b5c6 Simplify. Answer: LCM of Monomials and Polynomials Example 1A

= 240a3b5c6 Simplify. Answer: 240a3b5c6 LCM of Monomials and Polynomials = 240a3b5c6 Simplify. Answer: 240a3b5c6 Example 1A

B. Find the LCM of x3 – x2 – 2x and x2 – 4x + 4. LCM of Monomials and Polynomials B. Find the LCM of x3 – x2 – 2x and x2 – 4x + 4. x3 – x2 – 2x = x(x + 1)(x – 2) Factor the first polynomial. x2 – 4x + 4 = (x – 2)2 Factor the second polynomial. LCM = x(x + 1)(x – 2)2 Use each factor the greatest number of times it appears as a factor. Answer: Example 1B

B. Find the LCM of x3 – x2 – 2x and x2 – 4x + 4. LCM of Monomials and Polynomials B. Find the LCM of x3 – x2 – 2x and x2 – 4x + 4. x3 – x2 – 2x = x(x + 1)(x – 2) Factor the first polynomial. x2 – 4x + 4 = (x – 2)2 Factor the second polynomial. LCM = x(x + 1)(x – 2)2 Use each factor the greatest number of times it appears as a factor. Answer: x(x + 1)(x – 2)2 Example 1B

A. Find the LCM of 6x2zy3, 9x3y2z2, and 4x2z. A. x2z B. 36x2z C. 36x3y3z2 D. 36xyz Example 1A

A. Find the LCM of 6x2zy3, 9x3y2z2, and 4x2z. A. x2z B. 36x2z C. 36x3y3z2 D. 36xyz Example 1A

B. Find the LCM of x3 + 2x2 – 3x and x2 + 6x + 9. A. x(x + 3)2(x – 1) B. x(x + 3)(x – 1) C. x(x – 1) D. (x + 3)(x – 1) Example 1B

B. Find the LCM of x3 + 2x2 – 3x and x2 + 6x + 9. A. x(x + 3)2(x – 1) B. x(x + 3)(x – 1) C. x(x – 1) D. (x + 3)(x – 1) Example 1B

Concept

Simplify each numerator and denominator. Monomial Denominators Simplify . The LCD is 42a2b2. Simplify each numerator and denominator. Add the numerators. Example 2

Monomial Denominators Answer: Example 2

Monomial Denominators Answer: Example 2

Simplify . A. B. C. D. Example 2

Simplify . A. B. C. D. Example 2

Factor the denominators. Polynomial Denominators Simplify . Factor the denominators. The LCD is 6(x – 5). Subtract the numerators. Example 3

Distributive Property Polynomial Denominators Distributive Property Combine like terms. Simplify. Simplify. Answer: Example 3

Distributive Property Polynomial Denominators Distributive Property Combine like terms. Simplify. Simplify. Answer: Example 3

Simplify . A. B. C. D. Example 3

Simplify . A. B. C. D. Example 3

The LCD of the numerator is ab. The LCD of the denominator is b. Complex Fractions with Different LCDs Simplify . The LCD of the numerator is ab. The LCD of the denominator is b. Example 4

Simplify the numerator and denominator. Complex Fractions with Different LCDs Simplify the numerator and denominator. Write as a division expression. Multiply by the reciprocal of the divisor. Simplify. Example 4

Complex Fractions with Different LCDs Answer: Example 4

Complex Fractions with Different LCDs Answer: Example 4

Simplify . A. B. –1 C. D. Example 4

Simplify . A. B. –1 C. D. Example 4

The LCD of all of the denominators is xy. Multiply by Complex Fractions with Same LCD Simplify The LCD of all of the denominators is xy. Multiply by Distribute xy. Example 5

Complex Fractions with Same LCD Answer: Example 5

Complex Fractions with Same LCD Answer: Example 5

Simplify A. B. C. D. Example 5

Simplify A. B. C. D. Example 5

End of the Lesson