Unit III Factoring Section 3.3 Common Factors of a Polynomial

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Transcription:

Sections 3.3-3.8 Math 1201 1 Unit III Factoring Section 3.3 Common Factors of a Polynomial

Sections 3.3-3.8 Math 1201 2

Sections 3.3-3.8 Math 1201 3 3.5 Polynomials of the form x 2 + bx + c Goals: Practice Questions: Multiplying P.155 P.156 binomials #4, #8, #15b, #16 Factoring x 2 + bx +c trinomials

Sections 3.3-3.8 Math 1201 4 (I) Multiplying Binomials To multiple binomials we can use the following methods: Method 1: Use a rectangle diagram. (x - 3)(x + 1) Method 2: Use the distributive property. (x - 3)(x + 5)

Sections 3.3-3.8 Math 1201 5 Expand and simplify. (a) (x 4)(x + 2) (b) (8 b)(3 b) Expanding binomials by algebra tiles Which algebra tile model best represents the expansion of (x + 4)(x + 3)? A. B. C. D.

Sections 3.3-3.8 Math 1201 6 (II) Factoring x 2 + bx + c Trinomials To factor a trinomial such as: x 2 12x + 20 We will expand the binomials first Expand: (x 2)(x 10) (x 2)(x 10) and look at the four terms that form the trinomial above. Note: The two middle terms and ADD to give Also, these two numbers and MULTIPLY to give To factor a trinomial such as: x 2 12x + 20 into the product of two binomials, two conditions must be satisfied. The two numbers must MULTIPLY to give The two numbers must ADD to give So the trinomial below factors as: What number combination satisfies both conditions? x 2 12x + 20 = ANSWER: AND We can check by expanding ( )( ) to see if it generates the trinomial above.

Sections 3.3-3.8 Math 1201 7 Factor each trinomial (a) x 2 2x 8 (b) p 2 12p + 35 (II) Factoring x 2 + bx + c Trinomials continued Factoring Trinomials written in ascending order. Factor P.166 P.167 (a) 24 2q + q 2 (b) 40 + 3r r #4c, d #6(i) c #9, #11 #14a, c, e, g 2 Factoring a trinomial with a common factor.

Sections 3.3-3.8 Math 1201 8 Factor (a) 3t 2 + 3t + 36 Procedure: Identify and remove the G.C.F. first Factor the remaining trinomial (b) 4x 3 44x 2 + 120x (c) 6m 2 + 18m + 60 3.6 Factoring Trinomials of the Form ax 2 + bx + c Goals: Expanding binomials (ax + b)(cx + d) Factoring ax 2 + bx +c trinomials by Trial and Error P.167 (I) #14b, d, f, h #15 #19a, f #21 Expanding Binomials Algebraically Determine the area for the rectangle Expanding by F.O.I.L. (First Outside Inside Last) (II) Expanding Binomials Visually by Algebra Tiles

Sections 3.3-3.8 Math 1201 9 Expand (2x + 1)(x + 3) Summary: Dimensions Expanding binomials (ax + b)(cx + d) by algebra tiles or F.O.I.L. produces a trinomial of the form Dimensions Area = (III) Factoring Trinomials of the Form ax 2 + bx + c If we are given area and required to produce the dimensions of the rectangle then the previous process is reversed as we produce the factors of a trinomial ax 2 + bx + c. (A) Factoring Trinomials by Algebra Tiles Factor Using Algebra Tiles

Sections 3.3-3.8 Math 1201 10 Area = 2x 2 + 7x + 3 Dimensions Procedure: (i) Using algebra tiles construct a rectangle that represents the area 2x 2 + 7x + 3 Dimensions (ii) From the rectangle of part (i) determine the binomial dimensions by reading the edge length and edge width of the rectangle. Factored Form of 2x 2 + 7x + 3 = (B) Factoring Using Trial and Error Method Factor 2x 2 + 7x + 3 2x 2 + 7x + 3 Step I Step II State all pairs of factors for the first and last terms Determine what factors from the first term that must multiply off factors of the last term so that each result will add up to the middle term.

Sections 3.3-3.8 Math 1201 11 Step III Step IV Use the correct arrangement for multiplication in Step II to write the two binomials. Check the order of the arrangement to see if expansion would produce the given trinomial. Factor each of the following: (a) 6x 2 + x 2 (b) 4x 2-4x 15 (c) 5x 2-2x + 2 Practice Questions: P.177-178 #5b, d #9a, d #10b, e #13c, d, h, #15 d, e, f, g, h

Sections 3.3-3.8 Math 1201 12 3.6 continued Factoring Trinomials of the Form ax 2 + bx + c Goal: Factoring Trinomials ax 2 + bx + c where G.C.F is removed first Factor completely. 1. 12x 2 2x 4 Step I Identify the first = Step II Factor out the Step III Factor completely. Factor remaining 2. 15x 4 39x 3 18x 2 3. 12p 3 22p 2 + 20p

Sections 3.3-3.8 Math 1201 13 3.7 Multiplying Polynomials Goal: Multiplying polynomials by the distributive property Recall: Distributive Property a(b + c) = when multiplying polynomials, every term from the first polynomial must be multiplied off every term in the second polynomial Expand and simplify: 1. (3g + 2)(2g 1) 2. (2r + 5t) 2 3. (2h + 5)(h 2 + 3h 4) 4. ( 3f 2 + 3f 2)(4f 2 f 6) 5. (2c 3)(c + 5) + 3(c 3)( 3c + 1)

Sections 3.3-3.8 Math 1201 14 6. (3x + y 1)(2x 4) (3x + 2y) 2 7. Determine the expression (in expanded form) that represents the area of the shaded region. 3x + 2 x + 5 2x x + 6 Practice Questions: P.186 #4 #5a, d, e, f, #7a (i), (ii) #8a, d, #9c

Sections 3.3-3.8 Math 1201 15 3.8 Factoring Special Polynomials Goals: Identifying and factoring perfect square trinomials Factoring the Difference of Two Squares Factoring Trinomials in Two Variables (I) Factoring Perfect Square Trinomials Factor: 1. 4x 2 + 12x + 9 2. 4 20x + 25x 2 The relationship between the factors and the trinomial? Perfect Square Trinomials: Produce the binomial factors. First and last terms are square terms. Pattern for factoring: a 2 + 2ab + b 2 = ( )( ) OR ( ) 2 a 2 2ab + b 2 = ( )( ) OR ( ) 2

Sections 3.3-3.8 Math 1201 16 Identify and factor each perfect square trinomial? 1. 25x 2 + 60x + 36 2. 16x 2 24x + 9 3. 2x 2 + 6x 8 (II) Factoring The Difference of Two Squares Factor the following: 1. x 2 25 2. h 2 64 3. x 2 + 36 Difference of Two Squares: Consist of terms that are both square terms. The algebraic operation between the terms is. Pattern for factoring: a 2 b 2 = ( )( ) Factor each binomial: 1. 25 36x 2 2. 5x 4 80y 4

Sections 3.3-3.8 Math 1201 17 (III) Factoring Trinomials in Two Variables Factor 1. 2a 2 7ab + 3b 2 2. 10c 2 cd 2d 2 Practice Questions: P.194 #5 #6 #8 #13

Sections 3.3-3.8 Math 1201 18 Unit 3 Factoring Review Key Ideas: Expanding binomials by F.O.I.L For ANY FACTORING PROBLEM always look to remove a first Factoring the difference of two squares by pattern a 2 b 2 = ( )( ) Factoring perfect square trinomials: a 2 + 2ab + b 2 = ( )( ) or ( ) 2 a 2 2ab + b 2 = ( )( ) or ( ) 2 1. Expand (3x + 4y)(5x 2y) 2. Factor completely: (a) 18 + 7x x 2 (b) 2y 3 8y 2 42y (c) 36 49x 2 (d) 48y 2 75 (e) 15x 2 14xy 8y 2 (f) 18x 3 + 3x 2 36x (g) 4x 2 20xy + 25y 2

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