Two short calculations
For P(x)=x³−3x+7, division by x−2 leaves remainder P(2)=8−6+7=9. Evaluating at −2 would use the wrong divisor zero. For (x−2)(x+5)(x+1), the x² coefficient is −2+5+1=4. The constant product is −10, so returning the product would confuse two different coefficients.
Absolute value describes distance
The equation |x−3|=5 gives endpoints −2 and eight. Their separation is ten, even though the right endpoint is eight. Moving the center from three to seven moves both endpoints by four and leaves their distance unchanged. Keep a coordinate distinct from a distance between coordinates.
Count real roots without finding them
For x²−6x+c=0, the discriminant is 36−4c. With c=8 it is positive and there are two distinct real roots. With c=9 it is zero and there is one repeated root. With c=10 it is negative and there are no real roots. A repeated root counts once when the question says distinct.
Transfer and check
Find the x² coefficient in (x+2)(x−4)(x+3): it is one. Check by multiplying the first two factors and collecting x² after the final multiplication. Then find the separation of the roots of |x+1|=4: eight. The same shortcut should follow from explicitly finding −5 and three.
What is the remainder when P(x)=x³−3x+7 is divided by x−2?
- The remainder theorem uses P(2), the value at the zero of the divisor.
- Compute 2³−3(2)+7.
- The remainder is 9; no full long division is required.
Compare the other answer choices
Omit the constant term.
Evaluate at −2 instead of the zero of x−2.
Reverse the sign of the linear 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.
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