Factoring Polynomials The Diamond Method

See if you can discover a pattern in the diamonds below. 10 6 4 -8 5 2 2 3 -1 -4 -2 4 -5 7 5 2

Fill in the diamonds below 12 6 -21 -40 4 3 -2 -3 7 -3 -8 5 7 -5 4 -3

Fill in the diamonds below 24 -12 -20 28 2 12 -1 2 -10 -4 -7 12 14 11 -8 -11

Factoring Polynomials How do you factor a trinomial with leading coefficient of 1? Factor x2 -13x +36

Factor x2 -13x + 36 You can use a diamond... +36 -9 -4 -13 Write the last term here. +36 Now, find factors that will multiply to the top number, and add to the bottom number. -9 -4 -13 Write the middle coefficient here The factors are (x – 9)(x – 4)

Factor x2 – 3x – 40 –40 –3 –8 +5 The factors are (x – 8)(x + 5)

The factors are (x + 12)(x + 2) Factor x2 + 14x + 24 24 14 12 2 The factors are (x + 12)(x + 2)

The factors are (x + 12)(x - 1) Factor x2 + 11x - 12 -12 11 12 -1 The factors are (x + 12)(x - 1)

The factors are (x - 10)(x + 2) Factor x2 - 8x - 20 -20 -8 -10 2 The factors are (x - 10)(x + 2)

The factors are (x - 4)(x - 7) Factor x2 -11x + 28 28 -11 -4 -7 The factors are (x - 4)(x - 7)

How do you factor a trinomial whose leading coefficient is not 1? Factor 3x2 + 13x + 4 12 12 1 13 (x + 12)(x + 1) = x2 + 13x + 12 What happened?

(x + 12)(x + 1) The diamond method needs help when the leading coefficient is not equal to 1. We must use the fact that the leading coefficient is 3. (x + 12)(x + 1) 3 3

Now, reduce the fractions, if possible Now, reduce the fractions, if possible. The coefficient of x will be the reduced denominator. 12/3 = 4/1 1/3 is reduced. (1x + 4)(3x + 1)

Let’s try another. Factor 6x2 + x -15 -90 1 10 -9 (x + 10) (x – 9) , but we must divide. 6 6 Reduce the fractions.

10/6 = 5/3 -9/6 = -3/2 (3x + 5)(2x - 3) Now we’ll try an extra for experts factoring problem.

Now, factor the trinomial Factor 6d2 + 33d – 63 Remember, look for the GCF first... GCF: 3 3(2d2 + 11d – 21) Now, factor the trinomial -42 14 -3 11 Now we have the factors (x+14) (x-3)

Since the leading coefficient of our trinomial is 2, we need to divide by 2. (x + 14) (x - 3) Reduce 2 2 (x +7) (x - 3) = (1x + 7) (2x – 3) 1 2 Our complete factored form is 3(x + 7) (2x – 3)

Additional Practice

Practice 12 Factor the binomial x + 8x + 12 6 2 8 Did you get (x+6)(x+2). 2 Factor the binomial x - 6x + 5 5 Hint a negative sum and positive product means you are multiplying two negative numbers. -5 -1 -6 Did you get (x-5)(x-1).

Practice Factor the binomial x + 2x - 8 -8 4 -2 2 Hint a positive sum and negative product means you are multiplying a larger positive number by a smaller negative number. 4 -2 2 Did you get (x+4)(x-2). -10 2 Factor the binomial x - 3x - 10 -5 2 Hint a negative sum and negative product means you are multiplying a larger negative number by a smaller positive number. -3 Did you get (x-5)(x+2).

Practice Factor the binomial x - 9 -9 3 -3 Did you get (x+3)(x-3). 2 Factor the binomial x - 9 -9 Hint a zero sum and negative product means you are multiplying a positive number by an equal negative number. 3 -3 Did you get (x+3)(x-3). 2 This is called the difference of two squares. x is x times x and -9 is 3 times -3. The sum of 3x and -3x equals zero so there is no middle term. 2 (-x +3)(x+3) would give you -x + 9 which is also the difference of two squares (it could be written as 9- x ). 2

Factoring Completely 3 2 Factor the binomial 3x + 15x + 18x 2 x + 5x + 6 You could use a generic rectangle to factor out 3x. 3x Not so fast…. x2 + 5x + 6 can be further factored using a diamond. 6 3 2 Your answer should be 3x(x+3)(x+2) 5

Factoring Completely Now finish factoring 6 6 1 3 2 Factor the binomial 4x + 24x + 28x 2 x + 7x + 6 Did you could use a generic rectangle to factor out 4x. 4x Now finish factoring 6 6 1 Your answer should be 4x(x+6)(x+1) 7

Why The Diamond Method Works

Consider = acx2 + adx + bcx + bd = acx2 + (ad + bc)x + bd F O I L (ax + b)(cx +d) = acx2 + adx + bcx + bd = acx2 + (ad + bc)x + bd acbd ad + bc Now we have the factors (x + ad) (x + bc) bc ad

Since the leading coefficient of our trinomial is ac, we need to divide by ac. (x + bc) (x + ad) ac Reduce ac (x +b) (x - d) = (ax + b) (cx + d) a c