Factoring Trinomials ax Factoring Trinomials ax 22 + bx + c (a + bx + c (a > 1)> 1)

Model Model the following by the following by drawingdrawing rectangles with the given rectangles with the given areasareas..

LabelLabel the the sidessides.. FactorFactor the area/polynomial. the area/polynomial.

(Hint: Since the sides multiply to equal (Hint: Since the sides multiply to equal the area, the area, the sides are the factorsthe sides are the factors.).)

a)a) 2x 2x 2 2 + 7x + 6 + 7x + 6

Factoring Trinomials ax Factoring Trinomials ax 22 + bx + c (a + bx + c (a > 1)> 1)

Model Model the following by the following by drawingdrawing rectangles with the rectangles with the given given areasareas..

LabelLabel the the sidessides.. FactorFactor the area/polynomial. the area/polynomial. (Hint: Since the sides multiply to equal the area, (Hint: Since the sides multiply to equal the area, the the

sides are the factorssides are the factors.).)a)a) 2x 2x 2 2 + 7x + 6 + 7x + 6 2x 32x 3 x 2x x 2x 2 2 + 7x + 6 + 7x + 6 =(2x + 3)( x + 2)=(2x + 3)( x + 2) 22

Factoring Trinomials ax Factoring Trinomials ax 22 + bx + c (a + bx + c (a > 1)> 1)

b) 2x b) 2x 2 2 + 9x + 4 c) 3x + 9x + 4 c) 3x 22 + 11x + 6 + 11x + 6 d) 2x d) 2x 2 2 + 3x – 9 e) 2x + 3x – 9 e) 2x 2 2 – 7x + 3– 7x + 3f) 2x f) 2x 2 2 + 5x – 12 g) 3x + 5x – 12 g) 3x 22 - 16x + 16 - 16x + 16 h) 2x h) 2x 2 2 – 7x – 4 i) 3x – 7x – 4 i) 3x 22 - x - 4 - x - 4 j) 5x j) 5x 2 2 + x – 18 k) 3x + x – 18 k) 3x 22 - 4x - 15 - 4x - 15 l) 3x l) 3x 22 + 4x + 1 m) 4x + 4x + 1 m) 4x 2 2 + 4x – 15 + 4x – 15n) 2x n) 2x 2 2 – x – 1 o) 2x – x – 1 o) 2x 2 2 + 5x + 2 + 5x + 2p) 3x p) 3x 22 - 5x – 2 q) 3x - 5x – 2 q) 3x 22 - 4x + 1 - 4x + 1 r) r) Create your own trinomial of the form ax Create your own trinomial of the form ax 22 + bx + c that can be + bx + c that can be

factored. Factor. factored. Factor.

Factoring Trinomials ax Factoring Trinomials ax 22 + bx + c (a + bx + c (a > 1) - > 1) - ConclusionsConclusions

Have any conclusions/rules been Have any conclusions/rules been discovered about factoring ax discovered about factoring ax 22 + bx + c + bx + c other than drawing rectangles and other than drawing rectangles and determining the sides?determining the sides?

Ex. Ex. 2x 2x 2 2 + 5x – 12 + 5x – 12

Factoring Trinomials ax Factoring Trinomials ax 22 + bx + c (a + bx + c (a > 1) - > 1) - ConclusionsConclusions

Have any conclusions/rules been discovered Have any conclusions/rules been discovered about factoring ax about factoring ax 22 + bx + c + bx + c other than other than drawing rectangles and determining the sides?drawing rectangles and determining the sides?

Ex. Ex. 2x 2x 2 2 + 5x – 12 + 5x – 12 2x -32x -3 x 2x x 2x 2 2 + 5x – 12 + 5x – 12 = (2x - 3)( x + 4)= (2x - 3)( x + 4) 4 4 Have you noticed anything about the terms here?Have you noticed anything about the terms here?

x to = singles ( -3 x 4 = -12)x to = singles ( -3 x 4 = -12) and 2 x 4 + (-3) = 5and 2 x 4 + (-3) = 5

Factoring Trinomials ax Factoring Trinomials ax 22 + bx + c (a + bx + c (a > 1) - > 1) - ConclusionsConclusions

Although the previous observations are true they Although the previous observations are true they may not help that much. If applied to may not help that much. If applied to 2x 2x 2 2 + 9x + 4+ 9x + 4 it would require you to find two #’s that multiply it would require you to find two #’s that multiply to equal 4 while also finding a # that multiplies to equal 4 while also finding a # that multiplies by one of the factors used above plus the other by one of the factors used above plus the other to equal the 9.to equal the 9.

ex. 2x ex. 2x 2 2 + 9x + 4 + 9x + 4 (?x + ?)(x + ?)(?x + ?)(x + ?)

x to equal 4x to equal 4

Factoring Trinomials ax Factoring Trinomials ax 22 + bx + c (a + bx + c (a > 1) - > 1) - ConclusionsConclusions

Although the previous observations are true they may Although the previous observations are true they may not help that much. If applied to 2x not help that much. If applied to 2x 2 2 + 9x + 4 it would + 9x + 4 it would require you to find two #’s that multiply to equal 4 while require you to find two #’s that multiply to equal 4 while also finding a # that multiplies by one of the factors used also finding a # that multiplies by one of the factors used above plus the other to equal the 9.above plus the other to equal the 9.

ex. 2x ex. 2x 2 2 + 9x + 4 + 9x + 4 (?x + ?)(x + ?)(?x + ?)(x + ?)

x to equal 4x to equal 4 and x + = 9and x + = 9 For most of us, drawing or at least thinking about a For most of us, drawing or at least thinking about a

rectangle may be the best option.rectangle may be the best option.