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Algebra I Chapter 8

Algebra I Chapter 8

ProofsStandard 25.1

ProofsStandard 25.1 Students use properties of numbers to construct simple,valid arguments (direct and indirect) for, or formulate counterexamplesto, claimed assertions.When you solve an equation by factoring, you are using a deductive argument.Each step can be justified by an algebraic property.Solve 4x 2 - 324 = 0.4x 2 - 324 = 0Original equation(2x) 2 - 18 2 = 0 4x 2 = (2x) 2 and 324 = 18 2(2x + 18)(2x - 18) = 0 Factor the difference of squares.2x + 18 = 0 or 2x - 18 = 0Zero Product Propertyx = -9 x = 9 Solve each equation.Notice that the column on the left is a step-by-step process that leads to asolution. The column on the right contains the reasons for each statement.A two-column proof is a deductive argument that contains statements andreasons.Two-Column ProofGiven: a, x, and y are real numbers such that a ≠ 0, x ≠ 0, and y ≠ 0.Prove: ax 4 - ay 4 = a( x 2 + y 2 )(x + y)(x - y)There is a reason foreach statement.The first statementcontains the giveninformation.The last statementis what you wantto prove.Statements1. a, x, and y are real numbers such thata ≠ 0, x ≠ 0, and y ≠ 0.Reasons1. Given2. ax 4 - ay 4 = a( x 4 - y 4 ) 2. The GCF of ax 4 and ay 4 is a.3. ax 4 - ay 4 = a[ ( x 2 ) 2 - ( y 2 ) 2 ] 3. x 4 = ( x 2 ) 2 and y 4 = ( y 2 2)4. ax 4 - ay 4 = a( x 2 + y 2 )( x 2 - y 2 ) 4. Factor the difference of squares.5. ax 4 - ay 4 = a( x 2 + y 2 )(x + y)(x - y) 5. Factor the difference of squares.Reading to Learn1. Solve 1 _16 t 2 - 100 = 0 by using a two-column proof.2. Write a two-column proof using the following information. (Hint: Group terms withcommon factors.)Given: c and d are real numbers such that c ≠ 0 and d ≠ 0.Prove: c 3 - cd 2 - c 2 d + d 3 = (c + d)(c - d)(c - d)3. Explain how the process used to write two-column proofs can be useful in solvingFind the Error exercises, such as Exercise 37 on page 451.Reading Math Proofs 453

8-6Perfect Squares andFactoringMain Ideas• Factor perfect squaretrinomials.• Solve equationsinvolving perfectsquares.Standard 11.0Students applybasic factoringtechniques to second- andsimple third-degreepolynomials. Thesetechniques include finding acommon factor for all termsin a polynomial, recognizingthe difference of two squares,and recognizing perfectsquares of binomials.Standard 14.0 Studentssolve a quadratic equationby factoring or completingthe square. (Key)New Vocabularyperfect square trinomialsThe senior class has decided to build anoutdoor pavilion. It will have an 8-foot by8-foot portrayal of the school’s mascot in thecenter. The class is selling bricks with students’names on them to finance the project. If theysell enough bricks to cover 80 square feet andwant to arrange the bricks around the art, howwide should the border of bricks be?To solve this problem, you need to solve theequation (8 + 2x ) 2 = 144.Factor Perfect Square Trinomials Numbers like 16, 49, and 144 areperfect squares, since each can be expressed as the square of an integer.16 = 4 · 4 or 4 2 49 = 7 · 7 or 7 2 144 = 12 · 12 or 1 2 2Products of the form (a + b ) 2 and (a - b ) 2 , such as (8 + 2x ) 2 , are alsoperfect squares. Recall that these are special products that follow specificpatterns.(a + b ) 2 = (a + b)(a + b) (a - b ) 2 = (a - b)(a - b)x8 ftxxx8 ft= a 2 + ab + ab + b 2 = a 2 - ab - ab + b 2= a 2 + 2ab + b 2 = a 2 - 2ab + b 2These patterns can help you factor perfect square trinomials , which aretrinomials that are the squares of binomials.Squaring a Binomial(x + 7 ) 2 = x 2 + 2(x)(7) + 7 2= x 2 + 14x + 49(3x - 4 ) 2 = (3x) 2 - 2(3x)(4) + 4 2= 9 x 2 - 24x + 16Factoring a Perfect Squarex 2 + 14x + 49 = x 2 + 2(x)(7) + 7 2= (x + 7 ) 29 x 2 - 24x + 16 = (3x) 2 - 2(3x)(4) + 4 2= (3x - 4 ) 2For a trinomial to be factorable as a perfect square, three conditions mustbe satisfied as illustrated in the example below.❶The fi rst term mustbe a perfect square.4x 2 = (2x) 24 x 2 + 20x + 25❸The middle term must be twicethe product of the square rootsof the fi rst and last terms.2(2x)(5) = 20x❷The last term mustbe a perfect square.25 = 5 2454 Chapter 8 Factoring

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