Simplifying Expressions

Simplifying Expressions
08/09/12 lntaylor ©

Simplifying Fractions Simplifying Polynomials
Table of Contents Learning Objectives Simplifying Fractions Simplifying Polynomials Simplifying Rational Expressions The Distributive Property Practice 1 2 3 4 5 6 08/09/12 lntaylor ©

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

Definitions Definition 1 Expressions do not contain = ≠ < > ≤ ≥
Fractions are rational numbers consisting of a numerator and denominator i.e. ¼ , ½ , ¾ Definition 3 Terms are numbers, letters and exponents, or a combination of these things, separated by an operand symbol (+, −, ∗, ÷) Example 2x + 3 where 2x and 3 are both terms 3x2÷ x where 3x2 and x are both terms 08/09/12 lntaylor © TOC

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

Simplifying Fractions
Note1 The following is a review of the Fractions PowerPoint 08/09/12 lntaylor © TOC

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

Adding Fractions 08/09/12 lntaylor © TOC

2 2 + 3 3 5 5 7 7 + 15 = 29 + 14 5 x 7 = 35 35 Step 1 Construct matrix with numerators on top and denominators on side Step 2 Blank out boxes diagonally Step 3 Multiply matrix Step 4 Add the results; this becomes the numerator Step 5 Multiply left side numbers (denominators); this becomes the denominator Step 6 Reduce fraction if possible 08/09/12 lntaylor © TOC

Now you try 08/09/12 lntaylor © TOC

3 3 + 5 5 4 4 7 7 + 20 = 41 + 21 4 x 7 = 28 28 Step 1 Construct matrix with numerators on top and denominators on side Step 2 Blank out boxes diagonally Step 3 Multiply matrix Step 4 Add the results; this becomes the numerator Step 5 Multiply left side numbers (denominators); this becomes the denominator Step 6 Reduce fraction if possible 08/09/12 lntaylor © TOC

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

Alternate Method 08/09/12 lntaylor © TOC

3 3 + 5 5 ─ - 1 1 4 4 7 7 6 6 6 3 x 7 x 6 = 126 5 x 4 x 6 = 120 4 x 7 x 6 = 168 -1 x 7 x 4 = - 28 ___ 168 218 218 = = 84 218–168 = 50 Step 1 Multiply every numerator by every other denominator Step 2 Add the results; this is your numerator Step 3 Multiply the denominators; this is your denominator Step 4 Reduce fraction if possible Step 5 The easy way to reduce fractions is… Subtract the numerator and denominator… Do this until the result is less than the denominator and reduce 08/09/12 lntaylor © TOC

Now you try! 08/09/12 lntaylor © TOC

3 3 + 5 5 + 1 1 5 5 7 7 3 3 3 3 x 7 x 3 = 63 5 x 5 x 3 = 75 5 x 7 x 3 = 105 1 x 7 x 5 = 35 173 ___ 105 173 = = 105 173–105 = 68 Step 1 Multiply every numerator by every other denominator Step 2 Add the results; this is your numerator Step 3 Multiply the denominators; this is your denominator Step 4 Reduce fraction if possible Step 5 The easy way to reduce fractions is… Subtract the numerator and denominator… Do this until the result is less than the denominator and reduce 08/09/12 lntaylor © TOC

2 2 (5) 5 = 10 3 3 7 7 21 Rule 1 Multiply numerators; this becomes the new numerator Rule 2 Multiply denominators; this becomes the new denominator Rule 3 Reduce fraction if possible 08/09/12 lntaylor © TOC

Now you try! 3 3 4 4 08/09/12 lntaylor © TOC

3 3 (3) 3 = 9 4 4 4 4 16 Rule 1 Multiply numerators; this becomes the new numerator Rule 2 Multiply denominators; this becomes the new denominator Rule 3 Reduce fraction if possible 08/09/12 lntaylor © TOC

Understanding Cross Cancellation
3 7 4 6 08/09/12 lntaylor © TOC

3 3 (7) 7 1 1 x 7 = 7 (4) 4 6 6 2 2 x 4 = 8 Rule 1 Numerators can be moved anytime YOU want Rule 2 Reduce fraction Rule 3 Multiply straight across Rule 4 Reduce fraction if possible 08/09/12 lntaylor © TOC

Now you try! 3 5 4 9 08/09/12 lntaylor © TOC

3 3 (5) 5 1 1 x 5 = 5 (4) 4 9 9 3 3 x 4 = 12 Rule 1 Numerators can be moved anytime YOU want Rule 2 Reduce fraction Rule 3 Multiply straight across Rule 4 Reduce fraction if possible 08/09/12 lntaylor © TOC

Dividing Fractions 08/09/12 lntaylor © TOC

Divide 3 / 5 4 9 08/09/12 lntaylor © TOC

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

Now you try! 3 / 4 5 7 08/09/12 lntaylor © TOC

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

Simplifying Polynomials

Simplify a Polynomial Expression
3x2 + 3x x2 – 2x – 2 08/09/12 lntaylor © TOC

3x2 + 3x + 3 + x2 – 2x – 2 4x2 + x + 1 Step 1 Look for the same variable and exponent combinations Step 2 Combine like terms in columns Step 3 Add terms Note: When you add or subtract polynomials exponents do not change 08/09/12 lntaylor © TOC

Now you try 10x2 – 7x – 3x2 – 3x – 7 08/09/12 lntaylor © TOC

10x2 – 7x + 18 – 3x2 – 3x – 7 7x2 –10 x + 11 Step 1 Look for the same variable and exponent combinations Step 2 Combine like terms in columns Step 3 Add terms Note: When you add or subtract polynomials exponents do not change 08/09/12 lntaylor © TOC

Simplifying Rational Expressions

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

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

The Distributive Property

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

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

5(4x2 – 9x + 10) 5∗4x2 + 5∗-9x + 5∗10 20x2 – 45x + 50 Step 1
The Distributive Property means multiply the term outside the ( ) Step 2 Multiply coefficients and watch your signs Step 3 Rewrite with one sign for each term (not needed here) Step 4 Look for the same variable/exponent combinations (none here) Step 5 Combine any like terms in columns (none here) Note: When you add or subtract polynomials exponents do not change 08/09/12 lntaylor © TOC

– ( ) means a red flag – mistake zone
– 20x2 + 10x – 18 – 3(– 5x2 + 3x – 7) – – 15x2 – + 9x – – 21 + 15x2 – 9x + 21 – 5x2 + x + 3 – ( ) means a red flag – mistake zone Step 1 Multiply coefficients and then add the – to each sign in the ( ) Step 2 Step 3 Rewrite with one sign for each term Step 4 Look for the same variable and exponent combinations Step 5 Combine like terms in columns Step 6 Add terms Note: When you add or subtract polynomials exponents do not change 08/09/12 lntaylor © TOC

Now you try – 2x2 + 4x – 10 – 2(4x2 + 2x – 6)

– ( ) means a red flag – mistake zone
– 2x2 + 4x – 10 – 2(4x2 + 2x – 6) – + 8x2 – + 4x – – 12 – 8x2 – 4x + 12 – 10x2 + 2 – ( ) means a red flag – mistake zone Step 1 Multiply coefficients and then add the – to each sign in the ( ) Step 2 Step 3 Rewrite with one sign for each term Step 4 Look for the same variable and exponent combinations Step 5 Combine like terms in columns Step 6 Add terms Note: When you add or subtract polynomials exponents do not change 08/09/12 lntaylor © TOC

Simplify – 2x(– 3x + 8) – (2x + 9) Simplify 2(3/8 – 2/9)
Problem Answer Simplify 3/8 + 1/3 – 2/5 Simplify 2/3 – 5/6 + 1/2 Simplify 2x + 13 – 4x – 10 Simplify 2(– 3x – 7) Simplify 3x2(-3x – 7) Simplify – 2x(– 3x + 8) – (2x + 9) Simplify 2(3/8 – 2/9) Simplify 14x2 + 8x – 9 + 8x3 – 4x2 + 8 Simplify (2/3 + 1/9 – 1/3)2 > 37/120 > 1/3 > – 2x + 3 > – 6x – 14 > – 9x3 – 21x2 > 6x2 – 18x – 9 > 11/36 > 8x3 + 10x2 + 8x – 1 > 16/81 clear answers 08/09/12 lntaylor © TOC

Simplifying Expressions

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