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We are now going to try factoring a very specific type of polynomial, trinomials of the form 1x 2 + bx + c We are now going to try factoring a very specific type of polynomial, trinomials of the form 1x 2 + bx + c We are initially going to factor these trinomials the same way we factored at the beginning of the factoring unit and again when we learned how to common factor, by determining the sides of rectangles. We will model the trinomials we are trying to factor with our algebra tiles by constructing rectangles made up of the areas of the polynomials being factored and determine the factors by simply taking the sides of our rectangles. As we do, hopefully you will make connections and conclusions that will eventually allow you to factor trinomials without the tiles. We are initially going to factor these trinomials the same way we factored at the beginning of the factoring unit and again when we learned how to common factor, by determining the sides of rectangles. We will model the trinomials we are trying to factor with our algebra tiles by constructing rectangles made up of the areas of the polynomials being factored and determine the factors by simply taking the sides of our rectangles. As we do, hopefully you will make connections and conclusions that will eventually allow you to factor trinomials without the tiles.

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Factoring Trinomials 1x 2 + bx + c Factoring Trinomials 1x 2 + bx + c Model the following by drawing rectangles with the given areas. Model the following by drawing rectangles with the given areas. Label the sides. Label the sides. Factor the polynomial/area. Factor the polynomial/area. (Hint: since the sides of a rectangle multiply to equal the area, the sides are the factors) (Hint: since the sides of a rectangle multiply to equal the area, the sides are the factors) a) x 2 + 5x + 6

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Factoring Trinomials 1x 2 + bx + c Model the following by drawing rectangles with the given areas. Model the following by drawing rectangles with the given areas. Label the sides. Label the sides. Factor the polynomial/area. Factor the polynomial/area. (Hint: since the area of a rectangle is found by multiplying the sides, the sides are the factors) (Hint: since the area of a rectangle is found by multiplying the sides, the sides are the factors) a) x 2 + 5x + 6 x 3 x 3 x 2 + 5x + 6 x 2 + 5x + 6 x = (x+3)(x+2) x = (x+3)(x+2) 2

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

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Factoring Trinomials 1x 2 + bx + c - Conclusions When factoring trinomials of the form 1x 2 + bx + c it is really just a matter of finding the ?’s that appear in the following (since the x 2 is a must since all the trinomials are 1x 2 + bx + c ) When factoring trinomials of the form 1x 2 + bx + c it is really just a matter of finding the ?’s that appear in the following (since the x 2 is a must since all the trinomials are 1x 2 + bx + c ) 1x 2 + bx + c 1x 2 + bx + c x ? x ? x 1x 2 + bx + c x 1x 2 + bx + c =(x + ?)(x + ?) =(x + ?)(x + ?) ?

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Factoring Trinomials 1x 2 + bx + c - Conclusions Ex. x 2 + 7x + 12 Ex. x 2 + 7x + 12 x ? x ? x x 2 + 7x + 12 x x 2 + 7x + 12 ? =(x + ?)(x + ?) ? =(x + ?)(x + ?)

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Factoring Trinomials 1x 2 + bx + c - Conclusions x 2 + 7x + 12 x 2 + 7x + 12 x 4 x 4 x x 2 + 7x + 12 x x 2 + 7x + 12 3 =(x + 4)(x + 3) 3 =(x + 4)(x + 3)

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Factoring Trinomials 1x 2 + bx + c - Conclusions Explain what you know about those ?’s (the #’s that replace them) in terms of the rectangle and the trinomial being factored. Explain what you know about those ?’s (the #’s that replace them) in terms of the rectangle and the trinomial being factored. - The ?’s must multiply to equal the # of singles/the c (since the ?’s are the dimensions of the singles and multiplying the dimensions should give the total) - 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 beside the singles and combining the 2 groups must result in the total amount of x’s) 1x 2 + bx +c 1x 2 + bx +c =( x + ?)( x + ?) =( x + ?)( x + ?) multiply to = singles/c multiply to = singles/c add to = x’s/b add to = x’s/b

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Factoring Trinomials 1x 2 + bx + c - Conclusions Use the conclusions to try factoring the following without drawing. a) 1x 2 + 10x + 24 = (x + ?)(x + ?) What do you know about the ?’s ? x to = 24 + to = 10 = (x + 4)(x +6) b) 1x 2 - 4x – 12 = (x + ?)(x + ?) What do you know about the ?’s ? x to = 12 + to = - 4 = (x – 6)(x + 2)

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