Polynomials: Zeros, Coefficient Relations and the Division Algorithm
Polynomials means zeros, the relationship between zeros and coefficients, and the division algorithm - usually one 1-mark and one 3-mark question.
Zeros and coefficients
For p(x) = ax² + bx + c with zeros α and β: α + β = −b/a and αβ = c/a. For the cubic ax³ + bx² + cx + d with zeros α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a. A quadratic with given zeros is k[x² − (sum)x + product]. A polynomial of degree n has at most n zeros; graphically these are the x-coordinates where the curve cuts the x-axis.
Worked example
Find the zeros of x² − 5x + 6 and verify the relations. x² − 5x + 6 = (x − 2)(x − 3), so the zeros are 2 and 3. Sum = 5 = −(−5)/1 = −b/a; product = 6 = 6/1 = c/a. Both verified. Second type: if one zero of 2x² − 8x + k is 3, then 3 + β = 4 gives β = 1, and 3 × 1 = k/2, so k = 6.
Division algorithm
p(x) = g(x)·q(x) + r(x) with deg r < deg g. If r(x) = 0 then g(x) is a factor of p(x).
Traps that cost marks
- The sum of zeros is −b/a; dropping the minus sign is the commonest error in the chapter.
- Zeros are numbers, not factors - write 2 and 3, not (x − 2).
- In long division keep terms in descending order and insert missing powers with coefficient 0.
- "Find a quadratic polynomial" is answered up to a constant multiple k; the general form is safer than assuming k = 1.