The decision and its conditions
A polynomial can be organized by coefficients, factors or graph behavior, and each representation reveals different information. A factor x−a forces the value at a to vanish. Division connects a dividend, divisor, quotient and remainder, so multiplying back is a complete check. A root with even multiplicity can touch the horizontal axis without changing sign, whereas an odd multiplicity changes sign. The smallest degree consistent with stated multiplicities is their sum, but additional unseen roots can increase degree.
Worked example
Factor x³−27 as (x−3)(x²+3x+c). Find c. Use the difference-of-cubes identity. The quadratic factor ends with the square of the cube root. Here c=3²=9; multiplying cancels the intermediate terms.
A contrasting decision
Dividing x³+(2)x²+(-13)x−6 by x−3 gives a quadratic quotient. What is its x coefficient? The leading quotient term is x²; subtracting (x−3)x² leaves an x² coefficient 5. The next quotient term is 5x. The full quotient is x²+5x+2; multiplying by x−3 restores the dividend.
Diagnose a competing method
The first example’s tempting answer 3 follows this mistake: Use the cube root without squaring it. A difference of cubes needs a quadratic cofactor, not a difference of squares.
Transfer before revealing
If x−4 is a factor of P(x)=x³+kx+8, find k. Pause and identify the original restrictions. Solution: The factor theorem requires P(4)=0. 64+4k+8=0. Divide the isolated term by 4: k=-18.
Check the boundary of the method
A root with even multiplicity can touch the horizontal axis without changing sign, whereas an odd multiplicity changes sign. After the leading quotient term, subtract the whole divisor product before reading the next coefficient. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
Factor x³−27 as (x−3)(x²+3x+c). Find c.
- Use the difference-of-cubes identity.
- The quadratic factor ends with the square of the cube root.
- Here c=3²=9; multiplying cancels the intermediate terms.
Compare the other answer choices
Use the cube root without squaring it.
Use the original constant rather than its two-thirds power.
Give the last cofactor term a negative sign.
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 overview