• We are now going to try factoring a very specific We are now going to try factoring a very specific type of polynomial, trinomials of the form 1x type of polynomial, trinomials of the form 1x 2 2 + + bx + cbx + c

• We are initially going to factor these trinomials We are initially going to factor these trinomials the same way we factored at the beginning of the the same way we factored at the beginning of the factoring unit and again when we learned how to factoring unit and again when we learned how to common factor, by determining the sides of common factor, by determining the sides of rectangles. We will model the trinomials we are rectangles. We will model the trinomials we are trying to factor with our algebra tiles by trying to factor with our algebra tiles by constructing rectangles made up of the areas of constructing rectangles made up of the areas of the polynomials being factored and determine the the polynomials being factored and determine the factors by simply taking the sides of our factors by simply taking the sides of our rectangles. As we do, hopefully you will make rectangles. As we do, hopefully you will make connections and conclusions that will eventually connections and conclusions that will eventually allow you to factor trinomials without the tiles. allow you to factor trinomials without the tiles.

Factoring Trinomials 1x Factoring Trinomials 1x 22 + bx + + bx + cc

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

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

(Hint: since the sides of a rectangle multiply (Hint: since the sides of a rectangle multiply to equal the area, to equal the area, the sides are the factorsthe sides are the factors))

a)a) x x 2 2 + 5x + 6 + 5x + 6

Factoring Trinomials 1x Factoring Trinomials 1x 22 + bx + c + bx + c

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

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

(Hint: since the area of a rectangle is found by (Hint: since the area of a rectangle is found by multiplying the sides, multiplying the sides, the sides are the factorsthe sides are the factors))

a)a) x x 2 2 + 5x + 6+ 5x + 6

x 3x 3 x x 2 2 + 5x + 6+ 5x + 6 x = (x+3)(x+2)x = (x+3)(x+2)

22

Factoring Trinomials 1x Factoring Trinomials 1x 22 + bx + c + bx + c

b)b) x x 2 2 + 4x + 4 + 4x + 4c) x c) x 22 -1x – 6 -1x – 6d) x d) x 2 2 + 2x – 8 + 2x – 8e) x e) x 2 2 – 6x + 8 – 6x + 8f) x f) x 2 2 – 4 – 4g) x g) x 2 2 + 3x – 10 + 3x – 10h) x h) x 2 2 + 6x + 5 + 6x + 5i) x i) x 2 2 – 8x + 12 – 8x + 12j) x j) x 2 2 + 8x + 15 + 8x + 15k) x k) x 2 2 – 9 – 9l) x l) x 2 2 + 7x + 10 + 7x + 10m) x m) x 2 2 – 4x -12 – 4x -12

Factoring Trinomials 1x Factoring Trinomials 1x 22 + bx + c - + bx + c - ConclusionsConclusions

• When factoring trinomials of the form When factoring trinomials of the form 1x1x22+ + bx + cbx + c it is really just a matter of finding it is really just a matter of finding the ?’s that appear in the following (since the ?’s that appear in the following (since the the x x 22 is a must since all the trinomials are is a must since all the trinomials are 1x 1x 22 + bx + c + bx + c ) )

1x 1x 22 + bx + c + bx + c x ?x ?

xx 1x 1x 22 + bx + c + bx + c =(x + ?)(x + ?)=(x + ?)(x + ?) ??

Factoring Trinomials 1x Factoring Trinomials 1x 22 + bx + c - + bx + c - ConclusionsConclusions

• Ex. x Ex. x 22 + 7x + 12 + 7x + 12

x ?x ?

x x x x 22 + 7x + 12 + 7x + 12

? =(x + ?)(x + ?)? =(x + ?)(x + ?)

Factoring Trinomials 1x Factoring Trinomials 1x 22 + bx + c - + bx + c - Conclusions Conclusions

x x 22 + 7x + 12 + 7x + 12

x 4x 4

x x x x 22 + 7x + 12 + 7x + 12

3 =(x + 4)(x + 3)3 =(x + 4)(x + 3)

Factoring Trinomials 1x Factoring Trinomials 1x 22 + bx + c - + bx + c - Conclusions Conclusions

• Explain what you know about those ?’s (the #’s that replace Explain what you know about those ?’s (the #’s that replace them) in terms of the rectangle and the trinomial being them) in terms of the rectangle and the trinomial being factored.factored.

-- The ?’s must multiply to equal the # of singles/the c (since The ?’s must multiply to equal the # of singles/the c (since the ?’s are the dimensions of the singles and multiplying the ?’s are the dimensions of the singles and multiplying the dimensions should give the total)the dimensions should give the total)

- The ?’s must add to equal the x’s/the b (since the ?’s are The ?’s must add to equal the x’s/the b (since the ?’s are also the # of x’s standing above the singles and lying also the # of x’s standing above the singles and lying beside the singles and combining the 2 groups must result beside the singles and combining the 2 groups must result in the total amount of x’s)in the total amount of x’s)

1x 1x 22 + bx +c + bx +c =( x + ?)( x + ?)=( x + ?)( x + ?)

multiply to = singles/cmultiply to = singles/c add to = x’s/badd to = x’s/b

Factoring Trinomials 1x Factoring Trinomials 1x 22 + bx + c - + bx + c - Conclusions Conclusions • Use the conclusions to try factoring the following Use the conclusions to try factoring the following

without drawing.without drawing.a) 1x a) 1x 22 + 10x + 24 + 10x + 24 = (x + ?)(x + ?) = (x + ?)(x + ?) What do you know about the ?’s ?What do you know about the ?’s ? x to = 24 x to = 24 + to = 10 + to = 10 = (x + 4)(x +6) = (x + 4)(x +6) b) 1x b) 1x 2 2 - 4x – 12- 4x – 12 = (x + ?)(x + ?)= (x + ?)(x + ?) What do you know about the ?’s ?What do you know about the ?’s ? x to = 12x to = 12 + to = - 4+ to = - 4 = (x – 6)(x + 2)= (x – 6)(x + 2)