# Concept derivation guides

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Concept guide



## [Polynomial factorization over the rationals: verifying common factors, binomial products and differences of squares by comparing coefficients](/en/derivations/derivation-mko-factorization-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-factorization)

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.

Polynomial factorization over the rationals

derivation-mko-factorization-01





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