Question

A student uses the division shown to divide
–3x4 + 15x3 – x + 5 by x – 5, and concludes that x – 5 is not a factor of the polynomial.
es002-1.jpg
Describe two errors the student made. Is x – 5 a factor? Explain.

Answer 1

Two mistakes:

1. They omitted the missing x^2 term. The polynomial is -3x^4 + 15x^3 + 0x^2 − x + 5, so the coefficients for division must include a 0 for x^2.
2. They used the wrong value in synthetic division (or got the sign wrong). Dividing by x − 5 corresponds to using +5, not −5.

Correct check (synthetic division with 5):
Coefficients: -3, 15, 0, -1, 5
Bring down -3.
5·(−3)=−15 → 15+(−15)=0.
5·0=0 → 0+0=0.
5·0=0 → 0+(−1)=−1.
5·(−1)=−5 → −5+5=0 (remainder 0).

Quotient = −3x^3 + 0x^2 + 0x − 1 = −3x^3 − 1, remainder 0. So x − 5 is a factor. (You can also check p(5)=−3·5^4+15·5^3−5+5=0.)