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Polynomials 02/11/12lntaylor

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Table of Contents Learning Objectives Adding Polynomials Distributing Negative Signs Multiplying Polynomials Special Case Binomials Shortcuts to Multiplying Binomials Dividing Polynomials Factoring Polynomials 02/11/12lntaylor

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Adding Polynomials 02/11/12lntaylor TOC

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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 02/11/12lntaylor TOC

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Now you try 10x 2 7x + 18 3x 2 3x 7 02/11/12lntaylor TOC

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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 02/11/12lntaylor TOC

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Now you try x 2 7x 18 8x 2 3x 9 02/11/12lntaylor TOC

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8x 2 x 2 Step 1 Step 2 9x 2 7x 18 3x 9 10 x 27 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 02/11/12lntaylor TOC

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Distributing Negative Signs 02/11/12lntaylor TOC

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Distributing a Red Flag 10x 2 7x + 18 (3x 2 3x 7) 02/11/12lntaylor TOC

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10x 2 Step 4 Step 5 7x 2 7x+ 18 4x+ 25 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 Add the to each sign in the ( ) + 3x 2 3x 7 Step 3 Rewrite with one sign for each term 3x 2 + 3x+ 7 (3x 2 3x 7) 02/11/12lntaylor TOC

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Now you try 20x 2 + 10x 18 ( 5x 2 + 3x 7) 02/11/12lntaylor TOC

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20x 2 Step 4 Step 5 15x 2 + 10x 18 + 7x 11 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 Add the to each sign in the ( ) 5x 2 + 3x 7 Step 3 Rewrite with one sign for each term + 5x 2 3x + 7 ( 5x 2 + 3x 7) 02/11/12lntaylor TOC

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Now you try 12x 2 + 14x 10 (4x 2 2x 7) 02/11/12lntaylor TOC

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12 x 2 Step 4 Step 5 8x 2 + 14x 10 + 16x 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 Add the to each sign in the ( ) + 4x 2 2x 7 Step 3 Rewrite with one sign for each term 4x 2 + 2x + 7 (4x 2 2x 7) 02/11/12lntaylor TOC

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Multiplying Polynomials 02/11/12lntaylor TOC

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20x 2 Step 4 Step 5 5x 2 + 10x 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 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) 02/11/12lntaylor TOC

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Now you try 12x 2 + 14x 10 6(4x 2 2x 6) 02/11/12lntaylor TOC

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12x 2 Step 4 Step 5 12x 2 + 14x 10 + 26 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 and then add the to each sign in the ( ) + 24x 2 12x 36 Step 3 Rewrite with one sign for each term 24x 2 + 12x + 36 6(4x 2 2x 6) 02/11/12lntaylor TOC + 28 x

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Multiplying binomials and trinomials (3x 2 + 4x) (4x 3 6) 02/11/12lntaylor TOC

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(3x 2 + 4x)(4x 3 6) Step 1 Multiply 1 st term times everything in 2 nd parenthesis 3x 2 (Multiply coefficients and add exponents) 3x 2 (4x 3 ) =+ 12x 5 3x 2 (-6) = -18x 2 Step 2Multiply 2nd term times everything in 2 nd parenthesis 4x 4x (4x 3 ) = + 16x 4 4x (-6) =-24x Step 3Add terms with the same bases and exponents Step 4Rearrange terms in descending order of exponents 12x 5 + 16x 4 18x 2 24x 02/11/12lntaylor TOC

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Now you try (3x 2 + 4x) (5x 4 8) 02/11/12lntaylor TOC

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(3x 2 + 4x)(5x 4 8) Step 1 Multiply 1 st term times everything in 2 nd parenthesis 3x 2 (Multiply coefficients and add exponents) 3x 2 (5x 4 ) =+ 15x 6 3x 2 (-8) =-24x 2 Step 2Multiply 2nd term times everything in 2 nd parenthesis 4x 4x (5x 4 ) =+ 20x 5 4x (-8) = -32x Step 3Add terms with the same bases and exponents Step 4Rearrange terms in descending order of exponents 15x 6 + 20x 5 24x 2 32x 02/11/12lntaylor TOC

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Special Case Binomials 02/11/12lntaylor TOC

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Special Case Binomials Perfect square binomials (x + 4) 2 02/11/12lntaylor TOC

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(x + 4) 2 Step 1 Square the first term x 2 Step 2 Multiply last term by 2x (+ 4) 2x Step 3 Square the last term + 4 2 Step 4 Rewrite the terms x 2 + 8x + 16 02/11/12lntaylor TOC

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Now you try (x 8) 2 02/11/12lntaylor TOC

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(x 8) 2 Step 1 Square the first term x 2 Step 2 Multiply last term by 2x ( 8) 2x Step 3 Square the last term 8 2 Step 4 Rewrite the terms x 2 16x + 64 02/11/12lntaylor TOC

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Special Case Binomials Difference of squares binomials (x + 4) (x 4) 02/11/12lntaylor TOC

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Step 1 Square the first term (x + 4) (x 4)(x)x Step 2Multiply the last terms in each parenthesis ( + 4)( 4) Step 3 Rewrite x 2 - 16 Step 4 Note there are only two terms 02/11/12lntaylor TOC

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Now you try (x + 6) (x 6) 02/11/12lntaylor TOC

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Step 1 Square the first term (x + 6) (x 6)(x)x Step 2Multiply the last terms in each parenthesis ( + 6)( 6) Step 3 Rewrite x 2 - 36 Step 4 Note there are only two terms 02/11/12lntaylor TOC

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Shortcuts to Multiplying Binomials 02/11/12lntaylor TOC

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Shortcuts (x + 8) (x + 6) 02/11/12lntaylor TOC

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(x + 8) (x + 6) Step 1 Multiply the first terms in each parenthesis (x)x Step 2 Add the last terms in each parenthesis (+ 8+ 6) Step 3 Multiply by the variable!!!! x Step 4 Multiply the last terms in each parenthesis + 8(6) Step 5 Rewrite x 2 + 14x + 48 02/11/12lntaylor TOC

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Now you try (x + 10) (x 4) 02/11/12lntaylor TOC

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(x + 10) (x 4) Step 1 Multiply the first terms in each parenthesis (x)x Step 2 Add the last terms in each parenthesis (+ 10 4) Step 3 Multiply by the variable!!!! x Step 4 Multiply the last terms in each parenthesis + 10( 4) Step 5 Rewrite x 2 + 6x 40 02/11/12lntaylor TOC

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Now you try (x 8) (x 3) 02/11/12lntaylor TOC

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(x 8) (x 3) Step 1 Multiply the first terms in each parenthesis (x)x Step 2 Add the last terms in each parenthesis ( 8 3) Step 3 Multiply by the variable!!!! x Step 4 Multiply the last terms in each parenthesis 8( 3) Step 5 Rewrite x 2 11x + 24 02/11/12lntaylor TOC

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Dividing Polynomials 02/11/12lntaylor TOC

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Dividing Polynomials 4x 2 + 6x 2x 02/11/12lntaylor TOC

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4x 2 + 6x 2x Step 1 Split the terms 4x 2 2x + 6x 2x Step 2 Simplify: reduce the coefficients and subtract the exponents 2x + 3 02/11/12lntaylor TOC

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Now you try 14x 4 + 6x 2 3x 2x 02/11/12lntaylor TOC

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14x 4 + 6x 2 3x 2x Step 1 Split the terms 14x 4 2x + 6x 2 2x Step 2 Simplify: reduce the coefficients and subtract the exponents 7x 3 + 3x 3x 2x 3 2 02/11/12lntaylor TOC

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Factoring Polynomials 02/11/12lntaylor TOC

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Factoring Polynomials x 2 + 6x + 8 02/11/12lntaylor TOC

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x 2 + 6x + 8 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 + 6 2 = 3 Step 3 Square the answer and check last term (3)(3) = 9 8 Step 4 If they are both the same you have your factors No match Step 5 If they are not the same subtract one and add one (2)(4) = 8 Step 6 Continue until numbers match Yes Step 7 Add an x to each parenthesis (x + 2)(x + 4) 02/11/12lntaylor TOC

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Now you try x 2 20x + 96 02/11/12lntaylor TOC

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x 2 20x + 96 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 20 2 = 10 Step 3 Square the answer and check last term (-10)(-10) = 100 96 Step 4 If they are both the same you have your factors Step 5 If they are not the same subtract one and add one (-11)(-9) = 99 Step 6 Continue until numbers match Yes (-12)(-8) = 96 Step 7 Add an x to each parenthesis (x 12)(x 8) 02/11/12lntaylor TOC

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Now you try x 2 + 15x + 56 02/11/12lntaylor TOC

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x 2 + 15x + 56 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 + 15 2 = 7.5 Step 3 Since the answer includes 0.5 round up and down (8)(7) = 56 56 Step 4 If they are both the same you have your factors Step 5 If they are not the same subtract one and add one Step 6 Continue until numbers match Yes Step 7 Add an x to each parenthesis (x + 8)(x + 7) 02/11/12lntaylor TOC

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Now you try x 2 19x + 84 02/11/12lntaylor TOC

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x 2 19x + 84 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 19 2 = 9.5 Step 3 Since the answer includes 0.5 round up and down (-10)(-9) = 90 84 Step 4 If they are both the same you have your factors Step 5 If they are not the same subtract one and add one (-11)(-8) = 88 Step 6 Continue until numbers match Yes (-12)(-7) = 84 Step 7 Add an x to each parenthesis (x 12)(x 7) 02/11/12lntaylor TOC

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Factoring Negative Constants x 2 2x 48 02/11/12lntaylor TOC

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x 2 2x 48 Step 1 Check to see if last term is negative Step 2 Construct a table for the difference of factors for the constant 22422 31613 4128 682 Step 3 Match the middle term coefficient to the last column 2 Step 4 Use the numbers in the 1 st and 2 nd columns; add a variable to each (x 6) (x 8) Step 5 Put the middle term sign next to the largest number Step 6 Put the opposite sign next to the smallest number!!!! + 02/11/12lntaylor TOC

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Now you try x 2 11x 12 02/11/12lntaylor TOC

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x 2 11x 12 Step 1 Check to see if last term is negative Step 2 Construct a table for the difference of factors for the constant 11211 264 341 Step 3 Match the middle term coefficient to the last column 11 Step 4 Use the numbers in the 1 st and 2 nd columns; add a variable to each (x 1) (x 12) Step 5 Put the middle term sign next to the largest number Step 6 Put the opposite sign next to the smallest number!!!! + 02/11/12lntaylor TOC

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Special cases x 2 49 02/11/12lntaylor TOC

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x 2 49 Step 1 There are only two terms and the last term is negative Step 2 Square root the last term 49 = 7 Step 3 Write one factor with + and one factor with (x + 7) (x 7) 02/11/12lntaylor TOC

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Now you try x 2 225 02/11/12lntaylor TOC

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x 2 225 Step 1 There are only two terms and the last term is negative Step 2 Square root the last term 225 = 15 Step 3 Write one factor with + and one factor with (x + 15) (x 15) 02/11/12lntaylor TOC

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Super duper complicated problems x 2 7x + 12 x 4 02/11/12lntaylor TOC

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Super duper complicated problems Not!!!!! x 2 7x + 12 x 4 Step 1 These are not really complicated!!! Step 2 Check if constant in the denominator divides the constant in the numerator + 12 4 3= Step 3 Add an x to get the answer x 3 02/11/12lntaylor TOC

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Super duper complicated problem you can try x 2 2x 35 x + 5 02/11/12lntaylor TOC

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Super duper complicated problems Not!!!!! x 2 2x 35 x + 5 Step 1 These are not really complicated!!! Step 2 Check if constant in the denominator divides the constant in the numerator 35 + 5 7= Step 3 Add an x to get the answer x 7 02/11/12lntaylor TOC