Exercise 178 Page 116

1. Find a c. Since we have that a= 2 and c=-4, the value of a c is 2* (-4)=-8.
2. Find factors of a c.Since ac=-8, which is negative, we need factors of a c to have opposite signs — one positive and one negative — in order for the product to be negative. Since b=7, which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor, so that their sum is positive.

c|c|c|c 1^(st)Factor &2^(nd)Factor &Sum &Result - 2 &4 &-2 + 4 &2 - 1 & 8 & - 1 + 8 &7

3. Rewrite bx as two terms. Now that we know which factors are the ones to be used, we can rewrite bx as two terms. 2x^2+7x-4 ⇒ 2x^2 - 1x + 8x-4

Factor the Quadratic Trinomial

Here we have a quadratic trinomial of the form ax^2+bx+c, where |a| ≠ 1 and there are no common factors. To factor this expression, we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b. 2(4x^2+12x+5 ) ⇔ 2( 4x^2+12x+5 ) We have that a= 4, b=12, and c=5. There are now three steps we need to follow in order to rewrite the above expression.

1. Find a c. Since we have that a= 2 and c=5, the value of a c is 4* 5=20.
2. Find factors of a c. Since ac=20, which is positive, we need factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Since b=12, which is also positive, those factors will need to be positive so that their sum is positive.

c|c|c|c 1^(st)Factor &2^(nd)Factor &Sum &Result 1 &20 &1 + 20 &21 2 & 10 & 2 + ( 10) &12

3. Rewrite bx as two terms. Now that we know which factors are the ones to be used, we can rewrite bx as two terms. 2(4x^2+12x+5 ) ⇕ 2 ( 4x^2 + 2x + 10x+5 )