Chapter Test
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Try to look for a pair of numbers that when multiplied gives the constant term, and when added is equal to the linear term of the expression.
(x-8)(x-9)
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)
Distribute (x-9)
Distribute x
Distribute - 8
Subtract term
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!