The decision and its conditions
Expanding and factoring reveal different parts of an expression. In a product of two binomials, the middle coefficient comes from both cross products; a shared factor divides every term of a polynomial. A rational expression can be combined only after identifying a common denominator, and a denominator may exclude more than one input. Canceling a factor does not restore an originally excluded input. Test a proposed algebraic identity at a convenient allowed input to disprove a mistake, but remember that one successful substitution does not prove an identity for all inputs.
Worked example
A rectangle has algebraic side lengths 3x + 5 and x + 3. In its area expression, what is the coefficient of x? The coefficient of x comes from the two cross products. They are (3x)×3 and 5×x. The coefficient is 3×3+5=14.
A contrasting decision
The greatest common factor of 8x³ + 12x² is kx², where k is positive. Find k. Use the smaller variable power and the greatest common divisor of the coefficients. The coefficient GCF is gcd(8,12)=4. The full common factor is 4x², leaving 2x+3.
Diagnose a competing method
The first example’s tempting answer 15 follows this mistake: Report the constant area term. Expand only the two terms that contain exactly one factor of x.
Transfer before revealing
At x=5, evaluate 1/x + 3/(2x) exactly. Pause and identify the requested quantity. Solution: Use the common denominator 2x. The first numerator becomes two, so the combined expression is 5/(2x). At x=5, the exact value is 1/2.
Check the boundary of the method
Canceling a factor does not restore an originally excluded input. The question asks only for the numerical coefficient of the greatest common factor. Compare the answer with the question’s units and conditions. One correct example does not prove an identity or a statistical generalization.
A rectangle has algebraic side lengths 3x + 5 and x + 3. In its area expression, what is the coefficient of x?
- The coefficient of x comes from the two cross products.
- They are (3x)×3 and 5×x.
- The coefficient is 3×3+5=14.
Compare the other answer choices
Ignore the factor three in one cross product.
Report the constant area term.
Multiply cross coefficients rather than adding them.
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.
GED Mathematical Reasoning Assessment Guide for Educators