Factors hand over their zeros
Show on a big screenx² − 5x + 6 is the same as (x − 2)(x − 3). Do the brackets already show its zeros?
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(x − 2)(x − 3) = 0.(x − 2)(x − 3) = 0.x − 2 = 0→x = 2.x − 3 = 0→x = 3.- Zeros: 2 and 3.
The rule
Yes. Each bracket is a factor, one part of the product. Each factor shows one zero. A product is 0 only when one of its factors is 0. Use x = 2 in (x − 2)(x − 3). The first bracket is 0. The whole product is 0, whatever the second bracket is.
So (x − p)(x − q) has zeros p and q. Here p and q are any numbers.
To split the middle term, write it as two terms. Find two numbers that add to the number before x. They multiply to the number before x² times the last number.
| Step | Working |
|---|---|
| First × last | 3 × 2 = 6 |
| The pair | −6 and −1, adding to −7 |
| Split | 3x² − 6x − x + 2 |
| Group | 3x(x − 2) − 1(x − 2) |
| Brackets | (3x − 1)(x − 2) |
From 3x − 1 = 0, x = 1/3. From x − 2 = 0, x = 2. The zeros are x = 1/3 and x = 2. Check x = 2: 3 × 4 − 14 + 2 = 0.
A polynomial of degree 2 is a quadratic. Degree 2 means the highest power of x is 2.
Watch out!
(x + 3)(x + 4) does not have zeros 3 and 4. x + 3 is 0 at x = −3. The zeros are −3 and −4.
Try the lab to finish this lesson.