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Simplifying Expressions 08/09/12lntaylor ©

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Table of Contents Learning Objectives Simplifying Fractions Simplifying Polynomials Simplifying Rational Expressions The Distributive Property Practice /09/12lntaylor © 6

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Learning Objectives LO 1Understand the difference between expressions and equations TOC LO 2Correctly simplify expressions containing fractions and exponents LO 3Correctly use the principle of CLT – combine like terms 08/09/12lntaylor ©

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Definitions Definition 1Expressions do not contain = ≠ ≤ ≥ TOC Definition 2Fractions are rational numbers consisting of a numerator and denominator i.e. ¼, ½, ¾ Definition 3Terms are numbers, letters and exponents, or a combination of these things, separated by an operand symbol ( +, −, ∗, ÷) Example 08/09/12lntaylor © 2x + 3 where 2x and 3 are both terms 3x 2 ÷ x where 3x 2 and x are both terms

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Previous Knowledge PK 1 PK 2 Basic Operations and Properties Fractions PK 3Combining Like Terms TOC 08/09/12lntaylor © PK 4Exponent Rules

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Simplifying Fractions Note1The following is a review of the Fractions PowerPoint TOC 08/09/12lntaylor ©

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Rule 1 Rule 2 Adding and subtracting fractions requires cross multiplication Multiplying fractions requires straight across multiplication Rule 3Dividing requires flipping a fraction and multiplying straight across Rule 4Learn to “get rid” of fractions by turning expressions into equations Basic Rules of Fractions TOC 08/09/12lntaylor ©

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Adding Fractions TOC 08/09/12lntaylor ©

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Step 1 Step 2 Construct matrix with numerators on top and denominators on side Blank out boxes diagonally Step 3Multiply matrix Step 4Add the results; this becomes the numerator = 29 5 x 7 = Step 5Multiply left side numbers (denominators); this becomes the denominator 35 Step 6Reduce fraction if possible TOC 08/09/12lntaylor ©

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Now you try TOC 08/09/12lntaylor ©

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Step 1 Step 2 Construct matrix with numerators on top and denominators on side Blank out boxes diagonally Step 3Multiply matrix Step 4Add the results; this becomes the numerator = 41 4 x 7 = Step 5Multiply left side numbers (denominators); this becomes the denominator 28 Step 6Reduce fraction if possible TOC 08/09/12lntaylor ©

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Is there another method? TOC 08/09/12lntaylor ©

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Alternate Method TOC 08/09/12lntaylor ©

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Step 1 Step 2 Multiply every numerator by every other denominator Add the results; this is your numerator Step 3Multiply the denominators; this is your denominator Step Reduce fraction if possible ─ 1 6 TOC 08/09/12lntaylor © x 7 x 6 = x 4 x 6 = x 7 x 4 = x 7 x 6 = 168 ___ 168 Step 5The easy way to reduce fractions is… Subtract the numerator and denominator… Do this until the result is less than the denominator and reduce 218–168 = = =

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Now you try! TOC 08/09/12lntaylor ©

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Step 1 Step 2 Multiply every numerator by every other denominator Add the results; this is your numerator Step 3Multiply the denominators; this is your denominator Step Reduce fraction if possible TOC 08/09/12lntaylor © x 7 x 3 = x 5 x 3 = x 7 x 5 = x 7 x 3 = 105 ___ 105 Step 5The easy way to reduce fractions is… Subtract the numerator and denominator… Do this until the result is less than the denominator and reduce 173–105 = = =

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Rule 1 Rule 2 Adding and subtracting fractions requires cross multiplication Multiplying fractions requires straight across multiplication Rule 3Dividing requires flipping a fraction and multiplying straight across Rule 4Learn to “get rid” of fractions by turning expressions into equations Basic Rules of Fractions TOC 08/09/12lntaylor ©

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Rule 1 Rule 2 Multiply numerators; this becomes the new numerator Multiply denominators; this becomes the new denominator Rule 3Reduce fraction if possible (5)(5)= TOC 08/09/12lntaylor ©

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Now you try! TOC 08/09/12lntaylor ©

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Rule 1 Rule 2 Multiply numerators; this becomes the new numerator Multiply denominators; this becomes the new denominator Rule 3Reduce fraction if possible (3)(3)= TOC 08/09/12lntaylor ©

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Understanding Cross Cancellation TOC 08/09/12lntaylor ©

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Rule 1 Rule 2 Numerators can be moved anytime YOU want Reduce fraction Rule 3Multiply straight across (7)(7) (4) x 7 = 7 2 x 4 = 8 Rule 4Reduce fraction if possible TOC 08/09/12lntaylor ©

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Now you try! TOC 08/09/12lntaylor ©

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Rule 1 Rule 2 Numerators can be moved anytime YOU want Reduce fraction Rule 3Multiply straight across (5)(5) (4) x 5 = 5 3 x 4 = 12 Rule 4Reduce fraction if possible TOC 08/09/12lntaylor ©

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Dividing Fractions TOC 08/09/12lntaylor ©

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Rule 1 Rule 2 Adding and subtracting fractions requires cross multiplication Multiplying fractions requires straight across multiplication Rule 3Dividing requires flipping a fraction and multiplying straight across Rule 4Learn to “get rid” of fractions by turning expressions into equations Basic Rules of Fractions TOC 08/09/12lntaylor ©

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Divide / TOC 08/09/12lntaylor ©

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Rule 1 Rule 2 Write top fraction Flip bottom fraction Rule 3Check for cross cancellation; you can here but we will skip it Rule 4Multiply straight across ─ x 5 = 15 4 x 9 = 36 Rule 5Reduce fraction if possible 5 12 TOC 08/09/12lntaylor ©

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Now you try! / TOC 08/09/12lntaylor ©

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Rule 1 Rule 2 Write top fraction Flip bottom fraction Rule 3Check for cross cancellation; none here Rule 4Multiply straight across ─ x 7 = 21 5 x 4 = 40 Rule 5Reduce fraction if possible TOC 08/09/12lntaylor ©

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Simplifying Polynomials TOC 08/09/12lntaylor ©

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Simplify a Polynomial Expression 3x 2 + 3x x 2 – 2x – 2 TOC 08/09/12lntaylor ©

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+ x 2 3x 2 Step 1 Step 2 4x 2 + 3x+ 3 – 2x– 2 + x+ 1 Look for the same variable and exponent combinations Combine like terms in columns Step 3Add terms Note:When you add or subtract polynomials exponents do not change TOC 08/09/12lntaylor ©

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Now you try 10x 2 – 7x + 18 – 3x 2 – 3x – 7 TOC 08/09/12lntaylor ©

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– 3x 2 10x 2 Step 1 Step 2 7x 2 – 7x+ 18 – 3x– 7 –10 x+ 11 Look for the same variable and exponent combinations Combine like terms in columns Step 3Add terms Note:When you add or subtract polynomials exponents do not change TOC 08/09/12lntaylor ©

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Simplifying Rational Expressions TOC 08/09/12lntaylor ©

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Simplify 2x 2 + 4x – 10x 3 5 TOC 08/09/12lntaylor ©

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Step 4 Step 5 Combine like terms if necessary Divide by the y coefficient Step 6Simplify if possible Step 7You can erase the “= y ” if you want Step 2 Step 1 Turn the expression into an equation by introducing “ = y” Every term gets a denominator Step 3 Multiply every term’s numerator with every other denominator Then multiply the denominators 2x² 3 + 4x– 10x 1 5 =y (5)(1) 2x²+ 4x (3)(1) – 10x (3)(5)(1)(3)(5) = y 10x²+ 12x– 150x= 15y 10x² – 138x = 15y 10x² – 138x = y 15 x (10x – 138) 15 TOC 08/09/12lntaylor ©

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Now you try! 2x 2 + 3x – 10x 7 5 TOC 08/09/12lntaylor ©

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Step 4 Step 5 Combine like terms if necessary Divide by the y coefficient Step 6Simplify if possible Step 7You can erase the “= y ” if you want Step 2 Step 1 Turn the expression into an equation by introducing “ = y” Every term gets a denominator Step 3 Multiply every term’s numerator with every other denominator Then multiply the denominators 2x² 7 + 3x– 10x 1 5 =y (5)(1) 2x²+ 3x (7)(1) – 10x (7)(5)(1)(7)(5) = y 10x²+ 21x– 350x= 35y 10x² – 329x = 35y 10x² – 329x = y 35 x (10x – 329) 35 TOC 08/09/12lntaylor ©

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The Distributive Property TOC 08/09/12lntaylor ©

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Step 4 Step 5 Look for the same variable/exponent combinations (none here) Combine any like terms in columns (none here) Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 The Distributive Property means multiply the term outside the ( ) Multiply coefficients and watch your signs 3 ∗ 5x + 3 ∗ 7 Step 3 Rewrite with one sign for each term (not needed here) 3(5x + 7) TOC 08/09/12lntaylor © 15x + 21

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Now you try! 5(4x - 9) TOC 08/09/12lntaylor ©

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Step 4 Step 5 Look for the same variable/exponent combinations (none here) Combine any like terms in columns (none here) Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 The Distributive Property means multiply the term outside the ( ) Multiply coefficients and watch your signs 5 ∗ 4x + 5 ∗ -9 Step 3 Rewrite with one sign for each term (not needed here) 5(4x - 9) TOC 08/09/12lntaylor © 20x - 45

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Now you try! 5(4x 2 – 9x + 10) TOC 08/09/12lntaylor ©

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Step 4 Step 5 Look for the same variable/exponent combinations (none here) Combine any like terms in columns (none here) Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 The Distributive Property means multiply the term outside the ( ) Multiply coefficients and watch your signs 5 ∗ 4x ∗ -9x Step 3 Rewrite with one sign for each term (not needed here) 5(4x 2 – 9x + 10) TOC 08/09/12lntaylor © 20x 2 – 45x ∗ 10

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Now try something harder! – 20x 2 +10x – 18 – 3 (– 5x 2 + 3x – 7) TOC 08/09/12lntaylor ©

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– 20x 2 Step 4 Step 5 – 5x x– 18 + x + 3 Look for the same variable and exponent combinations Combine like terms in columns Step 6Add terms Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 – ( ) means a red flag – mistake zone Multiply coefficients and then add the – to each sign in the ( ) – – 15x 2 – + 9x– – 21 Step 3 Rewrite with one sign for each term + 15x 2 – 9x + 21 – 3(– 5x 2 + 3x – 7) TOC 08/09/12lntaylor ©

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Now you try – 2x 2 + 4x – 10 – 2(4x 2 + 2x – 6) TOC 08/09/12lntaylor ©

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– 2x 2 Step 4 Step 5 – 10x 2 + 4x– Look for the same variable and exponent combinations Combine like terms in columns Step 6Add terms Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 – ( ) means a red flag – mistake zone Multiply coefficients and then add the – to each sign in the ( ) – + 8x 2 – + 4x– – 12 Step 3 Rewrite with one sign for each term – 8x 2 – 4x + 12 – 2(4x 2 + 2x – 6) TOC 08/09/12lntaylor ©

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Practice TOC 08/09/12lntaylor ©

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08/09/12lntaylor © TOC ProblemAnswer Simplify 3/8 + 1/3 – 2/5 Simplify 2/3 – 5/6 + 1/2 Simplify 2x + 13 – 4x – 10 Simplify 2(– 3x – 7) Simplify 3x 2 (-3x – 7) Simplify – 2x(– 3x + 8) – (2x + 9) Simplify 2(3/8 – 2/9) Simplify 14x 2 + 8x – 9 + 8x 3 – 4x Simplify (2/3 + 1/9 – 1/3) 2 > 37/120 > > > > > 1/3 – 2x + 3 – 6x – 14 – 9x 3 – 21x 2 6x 2 – 18x – 9 > 11/36 > 8x x 2 + 8x – 1 > 16/81 clear answers

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