Unit III Factoring Section 3.3 Common Factors of a Polynomial

Transcription

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

2 Sections Math

3 Sections Math 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

4 Sections Math (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)

5 Sections Math 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.

6 Sections Math (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.

7 Sections Math 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) r r #4c, d #6(i) c #9, #11 #14a, c, e, g 2 Factoring a trinomial with a common factor.

8 Sections Math Factor (a) 3t 2 + 3t + 36 Procedure: Identify and remove the G.C.F. first Factor the remaining trinomial (b) 4x 3 44x x (c) 6m m 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

9 Sections Math 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

10 Sections Math 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.

11 Sections Math 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 #5b, d #9a, d #10b, e #13c, d, h, #15 d, e, f, g, h

12 Sections Math 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 x 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 p 3 22p p

13 Sections Math 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)

14 Sections Math (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

15 Sections Math 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 x x + 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

16 Sections Math Identify and factor each perfect square trinomial? 1. 25x x x 2 24x x 2 + 6x 8 (II) Factoring The Difference of Two Squares Factor the following: 1. x h x 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: x x 4 80y 4

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

18 Sections Math 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) x 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