1.1 Roots of polynomial equations

For ax²+bx+c=0 with roots α and β, α+β=−b/a and αβ=c/a. These relations are often faster and safer than applying the quadratic formula twice.

For higher-degree polynomials, factorisation and the factor theorem identify roots; repeated roots occur when a factor is squared or the polynomial and its derivative share a root. Keep the equation in standard form before reading coefficients.

For 2x²−5x+3=0, the roots have sum 5/2 and product 3/2, so their average is 5/4 without finding either root.

The coefficient relationships depend on signs and the leading coefficient; they are not simply “sum=b, product=c”.

If α is a root of f(x)=0, a new equation with roots such as α+k, kα or 1/α is obtained by substituting the inverse transformation into f, then simplifying.

For a shift y=x−k, replace x by y+k; for a scale y=x/k, replace x by ky. Check whether the question asks for roots or for an equation whose roots are transformed values.

If α satisfies α²−3α+2=0, the roots α+1 satisfy (y−1)²−3(y−1)+2=0, which simplifies to y²−5y+6=0.

Adding k to every root does not mean adding k to every coefficient, and reciprocal roots require multiplying by a suitable power to remove denominators.