Exercise 16 Page 539

To factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term. x^2+17x+72 In this case we have 72. This is a positive number, so for the product of the constant terms in the factors to be positive these constants must have the same sign (both positive or both negative).

Factor Constants	Product of Constants
1 and 72	72
-1 and -72	72
2 and 36	72
-2 and -36	72
3 and 24	72
-3 and -24	72
4 and 18	72
-4 and -18	72
6 and 12	72
-6 and -12	72
8 and 9	72
-8 and -9	72

Next, let's consider the coefficient of the linear term. x^2+17x+72 For this term we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 17. Since 17 > 0, we will be considering sums of postive numbers only.

Factors	Sum of Factors
1 and 72	73
2 and 36	38
3 and 24	27
4 and 18	22
6 and 12	18
8 and 9	17

We found the factors whose product is 72 and whose sum is 17. x^2+17x+72 ⇔ (x-8)(x-9)

Checking Our Answer

Check your answer ✓

We can check our answer by applying the Distributive Property and comparing the result with the given expression.

(x-8) (x-9)

Distr

Distribute (x-9)

x (x-9) - 8 (x-9)

Distr

Distribute x

x^2-9x - 8 (x-9)

Distr

Distribute - 8

x^2-9x-8x+72

SubTerm

Subtract term

x^2-17x+72

After applying the Distributive Property and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!