Exercise 10 Page 217

To factor a trinomial with a leading coefficient of one, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term. x^2-10x+21 In this case, we have 21. 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 21	21
- 1 and - 21	21
3 and 7	21
- 3 and - 7	21

Next, let's consider the coefficient of the linear term. x^2-10x+21 For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, - 10.

Factors	Sum of Factors
1 and 21	22
- 1 and - 21	- 22
3 and 7	10
- 3 and - 7	- 10

We found the factors whose product is 21 and whose sum is - 10. x^2-10x+21 ⇔ (x-3)(x-7)

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-3)(x-7)

Distr

Distribute x-7

x(x-7)-3(x-7)

Distr

Distribute x

x^2-7x-3(x-7)

Distr

Distribute - 3

x^2-7x-3x+21

SubTerm

Subtract term

x^2-10x+21

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!