The Zero Test: Factor Theorem
If p(a) = 0, then (x - a) is a factor. Verify it by expanding.
Equation
y = (x - 3)*(x + 2) - (x^2 + 0*x - 3*2)
Graph
Table
| x | y |
|---|---|
| -5 | 5 |
| -4 | 4 |
| -3 | 3 |
| -2 | 2 |
| -1 | 1 |
| 0 | 0 |
| 1 | -1 |
| 2 | -2 |
| 3 | -3 |
| 4 | -4 |
| 5 | -5 |
What this lesson covers
What you do
You shape the function y = (x - a)*(x + b) - (x^2 + k*x - a*b) and watch the graph answer.
Challenges to clear
- Expand (x−3)(x+2): keep a = 3, b = 2 and slide k to the middle coefficient (b − a). Difference 0 at x = 2.
- 0 at x = −1 too: if p(3) = 0 then (x − 3) is a factor — that is the zero test.
- Expand (x − 5)(x + 2): set a = 5 and b = 2.
- Now the middle coefficient — it is b − a, which goes negative here. Slide k until the difference is 0 at x = 2, and p(5) = 0 confirms (x − 5) is a factor.
Check yourself
Why does the graph staying flat at y=0 prove the identity?
If p(a) = 0 for a polynomial p(x), what does the Factor Theorem state?
Using the sliders, if a=3 and b=2, what is the expanded form of (x-3)(x+2)?
- It shows the difference between the factored form and expanded form is always zero. — correct
- It shows that x=0 is always a root of the polynomial.
- It proves that a and b must be equal for the equation to work.
- It indicates that the function has no real solutions.
- (x - a) is a factor of p(x). — correct
- (x + a) is a factor of p(x).
- a is the y-intercept of p(x).
- p(x) has no remainder when divided by x.
- x^2 - x - 6 — correct
- x^2 + x - 6
- x^2 - 5x - 6
- x^2 + 5x - 6
Think about it
- (x−3)(x+2) = x² + ?·x − 6. The middle coefficient is 2 − 3. What is it?