Polynomials
Polynomials tests degree, zeros, coefficient relations, division algorithm and graphs. BSEB often asks you to verify relations or find a quadratic from given zeros.
Core rules
- Degree is the highest power; linear, quadratic and cubic polynomials have degrees 1, 2 and 3.
- A zero of p(x) is a number c with p(c)=0.
- For ax²+bx+c, sum of zeros α+β=-b/a and product αβ=c/a.
- For ax³+bx²+cx+d, α+β+γ=-b/a, αβ+βγ+γα=c/a and αβγ=-d/a.
- Division algorithm: p(x)=g(x)q(x)+r(x), with r=0 or deg r < deg g.
- The graph of a quadratic is a parabola; its zeros are the x-intercepts.
Worked example
Find a quadratic polynomial with zeros 3 and -2. Sum = 3 + (-2) = 1; product = 3 × (-2) = -6. Using x²-(sum)x+product gives x²-x-6. Check: (x-3)(x+2)=x²-x-6, so the zeros are 3 and -2.
Traps
Sign errors are common: the sum of zeros is -b/a, not b/a. In “find polynomial”, keep the leading coefficient 1 unless a multiplier is required. The zero polynomial has no defined degree. In division, the remainder’s degree must be less than the divisor’s degree; stop when it is. Do not confuse p(0), the constant term, with a zero of the polynomial.