Concept guide
Polynomial factorization over the rationals: verifying common factors, binomial products and differences of squares by comparing coefficients
In ℚ[x]: (1) factor p(x)=3x²+3x−18 as 3(x+3)(x−2); (2) factor q(x)=2x²−50 as 2(x−5)(x+5); verify each factorization by expanding and comparing every coefficient. (3) Prove that x²+25≠(x−5)(x+5) and that x²+25 is not a product of two degree-one factors. (4) Explain why x³ and x take equal values at x=−1, 0, 1 and still are different polynomials.