Exercise 19 Page 521

We have a quadratic trinomial of the form ad^2+bd+c, where d is the variable and the absolute value of the leading coefficient a is different from 1. Since there are no common factors, we will rewrite the linear term - 4d as the sum of two linear terms. The coefficients of these two terms will be factors of ac. 4d^2-4d-35 ⇔ 4d^2+(- 4)d+(-35) We have that a= 4, b=- 4, and c=-35. 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= 4 and c=-35, the value of a c is 4* (-35)=-140.
2. Find factors of a c. Since ac=-140, which is negative, we need factors of a c to have opposite signs in order for the product to be negative. Since b=-4, which is also negative, the absolute value of the negative factor will need to be greater than the absolute value of the positive factor, so that their sum is negative.

c|c|c|c 1^(st)Factor &2^(nd)Factor &Sum &Result 1 &- 140 &1 + (-140) &- 139 2 &- 70 &2 + (-70) &- 68 4 &- 35 &4 + (-35) &- 31 5 &- 28 &5 + (-28) &- 23 10 & - 14 & 10 + ( - 14) &- 4

3. Rewrite bd as two terms. Now that we know which factors are the ones to be used, we can rewrite bd as two terms. 4d^2+(- 4)d-35 ⇔ 4d^2 - 14d+ 10d-35

Finally, we will factor the last expression obtained.