Math · College algebra · Concept
Factoring and zeros
Rewrite a polynomial as a product, then use its factors to understand where its graph meets the axis.
Why change the form?
Suppose you want to know where a quadratic graph crosses the horizontal axis. Expanded form tells you its coefficients, but the zeros can be hard to see. Factoring writes the same expression as a product, which makes those zeros easier to find.
Read the two conditions together
For a quadratic whose x² coefficient is 1, multiplying (x + p)(x + q) gives x² + (p + q)x + pq. So p and q must satisfy two conditions: their product is the constant term and their sum is the coefficient of x. Here we need a product of 6 and a sum of −5. The pair −2 and −3 does both.
| p | q | Product | Sum |
|---|---|---|---|
| 1 | 6 | 6 | 7 |
| 2 | 3 | 6 | 5 |
| −1 | −6 | 6 | −7 |
| −2 | −3 | 6 | −5 |
Use the zero-product property only after setting the expression equal to zero
A product is zero when at least one factor is zero. That lets us turn one quadratic equation into two simple possibilities. Notice the difference between factoring an expression and solving an equation: we need the equation to know what the expression equals.
Why can we set each factor equal to zero?
If both factors were nonzero real numbers, their product would also be nonzero. At least one must therefore be zero. This reasoning depends on the right-hand side being zero; you cannot use it directly on (x − 2)(x − 3) = 10.
Check by multiplying back
Multiply the factors before relying on them. The outer and inner products are −3x and −2x; together they make −5x. Checking only the first and last terms could let a sign error slip through.
Connect the factors to the graph
For y = x² − 5x + 6, the zeros give the points (2, 0) and (3, 0). Between them, one factor is positive and one is negative, so the graph lies below the axis. Outside that interval the factors have the same sign. This gives you a useful prediction before you open Graph.
What if no integer pair works?
A quadratic can have real roots without factoring neatly over the integers. Completing the square or using the quadratic formula handles those cases. Failure to find an integer pair is not proof that the equation has no solution.
Common mistakes
- Checking the constant term but forgetting the middle term.
- Dropping one root after setting a product equal to zero.
- Assuming every quadratic factors over the integers.
Work through an example
Solve x² − 5x + 6 = 0, then connect the answer to a graph.
Find both zeros of a quadratic →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.