A reliable method
Distribute products systematically and group terms with equal exponents. To evaluate a polynomial, substitute with parentheses and calculate powers before multiplying coefficients.
The common trap
A minus sign before parentheses changes every term inside. Missing one sign can change a constant, a coefficient and the resulting roots.
Check and transfer
The remainder when dividing a polynomial by x − a is its value at a. This allows a numerical remainder check without carrying out a full long division.
Worked comparison: identify the structure
Expand (x+2)(x+5)=x²+7x+10. The x coefficient comes from 5x and 2x; ten is the constant. A request for the coefficient of x is not a request to evaluate the expression at x=1.
Transfer and explain
For P(x)=x²−3x+5, the remainder after division by x−2 is P(2)=4−6+5=3. Evaluating at −2 would answer the different divisor x+2.
In the expansion of (x + 4)(x + 3), what is the coefficient of x?
- The x terms come from the two cross products.
- Add their coefficients while keeping the x² and constant terms separate.
- The requested value is 7. Substitute or reverse the operation to check this result.
Compare the other answer choices
Report the leading coefficient of x squared.
Keep only the second cross term.
Keep only the first cross term.
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