Chapter at a Glance

Here is the whole of Polynomials on one page — perfect for the night before your exam.

Basics

  • A polynomial has only non-negative integer powers of the variable. 1x\dfrac{1}{x}, x\sqrt{x} are NOT allowed.
  • Degree = highest power present. Linear (1), Quadratic (2), Cubic (3).
  • A zero of p(x)p(x) is a value kk with p(k)=0p(k) = 0.
  • A polynomial of degree nn has at most nn zeroes.

Remember: Zeroes of p(x)p(x) are the x-coordinates where the graph y=p(x)y = p(x) meets the x-axis.

Graphs and Zeroes

Polynomial Graph shape Max real zeroes
Linear ax+bax+b Straight line 1
Quadratic ax2+bx+cax^2+bx+c Parabola 2
Cubic ax3+…ax^3+\dots Curve with up to 2 bends 3
  • Parabola opens up if a>0a > 0, down if a<0a < 0.
  • Cuts x-axis at 2 points → 2 zeroes; touches → equal zeroes; misses → no real zeroes.

Key Point: Number of real zeroes = number of points where the graph meets the x-axis.

Zero–Coefficient Relations (Must Memorise)

Quadratic ax2+bx+cax^2 + bx + c (zeroes α,β\alpha, \beta)

α+β=−ba,αβ=ca\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}

Cubic ax3+bx2+cx+dax^3 + bx^2 + cx + d (zeroes α,β,γ\alpha, \beta, \gamma)

α+β+γ=−ba,αβ+βγ+γα=ca,αβγ=−da\alpha+\beta+\gamma = -\frac{b}{a}, \quad \alpha\beta+\beta\gamma+\gamma\alpha = \frac{c}{a}, \quad \alpha\beta\gamma = -\frac{d}{a}

Key Point: Signs alternate −,+,−-, +, -. The sum of zeroes always carries the minus sign.

Useful identities

  • α2+β2=(α+β)2−2αβ\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta
  • (α−β)2=(α+β)2−4αβ(\alpha - \beta)^2 = (\alpha+\beta)^2 - 4\alpha\beta
  • 1α+1β=α+βαβ\dfrac{1}{\alpha} + \dfrac{1}{\beta} = \dfrac{\alpha+\beta}{\alpha\beta}
  • α3+β3=(α+β)3−3αβ(α+β)\alpha^3 + \beta^3 = (\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta)

Forming Polynomials and Division

Forming from zeroes

  • Quadratic: x2−(sum)x+(product)x^2 - (\text{sum})x + (\text{product}).
  • Cubic: x3−S1x2+S2x−S3x^3 - S_1 x^2 + S_2 x - S_3, where S1,S2,S3S_1, S_2, S_3 are the sum, pair-sum, and product of zeroes.

Division algorithm

p(x)=g(x) q(x)+r(x),r(x)=0 or deg⁡r<deg⁡gp(x) = g(x)\,q(x) + r(x), \quad r(x) = 0 \text{ or } \deg r < \deg g

  • deg⁡q=deg⁡p−deg⁡g\deg q = \deg p - \deg g.
  • Remainder Theorem: remainder on dividing by (x−a)(x-a) is p(a)p(a).
  • Factor Theorem: (x−a)(x-a) is a factor ⇔ p(a)=0p(a) = 0.

Key Point: To find all zeroes when some are known, divide by the corresponding factor and solve the quotient.

Last-Minute Tips and Common Traps

  • The sum of zeroes is −b/a-b/a — don't forget the minus sign (the single most common error).
  • A pair of zeroes ±k\pm\sqrt{k} gives the factor x2−kx^2 - k; use it to reduce higher-degree polynomials.
  • 'Touching' the x-axis ⇒ equal (repeated) zeroes, not 'no zeroes'.
  • For 'form a polynomial from zeroes', remember the template signs: x2−Sx+Px^2 - Sx + P.
  • When dividing, always arrange in decreasing powers and insert 0x0x for missing terms.
  • Verify division with: dividend = divisor × quotient + remainder.

Final Word: Polynomials is a scoring chapter. Master the zero–coefficient relations, the forming templates, and polynomial division, and you secure most of the marks. All the best!