A reliable method
First remove a greatest common factor if one exists. Then look for a difference of squares or a trinomial whose product and sum conditions can be satisfied.
The common trap
The factors of an expression are not automatically solutions of an equation. The zero-product rule requires the expression to be set equal to zero.
Check and transfer
Expand a proposed factorization to check it. This verifies the middle term as well as the leading and constant terms, where an attractive-looking factor pair can still fail.
Worked comparison: identify the structure
The product (2x+1)(x−3) expands to 2x²−5x−3. The middle coefficient is −6+1, not 2−3 or the product of those cross coefficients. Expand before trusting a factorization that has the right constant term.
Transfer and explain
Set the product equal to zero: roots −1/2 and 3. Their sum is 5/2 and product −3/2. Contrast a factor constant, a root, and a combined quantity before deciding which answer to report.
Factor 8x² + 12x as 4x(ax + b). What is b?
- Divide the linear term by the factored-out term.
- The x cancels and the remaining coefficient is b.
- The requested value is 3. Substitute or reverse the operation to check this result.
Compare the other answer choices
Report a, the other coefficient inside the parentheses.
Leave the original linear coefficient unchanged after factoring.
Reverse the division used to recover b.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overviewSiyavula Grade 10 foundation reference