Polynomials — CDS Elementary Mathematics

Formulas (2)

• Degree, identities and integer values · Degree of a product
  $$\deg(fg) = \deg f + \deg g$$
• Zeros and coefficients · Cubic: sum of zeros
  $$\alpha + \beta + \gamma = -\dfrac ba, \quad \alpha\beta + \beta\gamma + \gamma\alpha = \dfrac ca, \quad \alpha\beta\gamma = -\dfrac da$$

Watch out for (2)

• Equal degrees can cancel
  → Degree, identities and integer values
• Divide by a
  → Zeros and coefficients

Formulas (3)

• Remainder on division by a linear factor · Remainder theorem
  $$f(x) \div (x - a) \;\Rightarrow\; \text{remainder} = f(a)$$
• Remainder on division by a quadratic · Linear remainder
  $$r(x) = \dfrac{(x - \beta)f(\alpha) - (x - \alpha)f(\beta)}{\alpha - \beta}$$
• Divisibility of xⁿ ± aⁿ · Factor of xⁿ − aⁿ
  $$x^n - a^n = (x - a)\left(x^{n - 1} + x^{n - 2}a + \cdots + a^{n - 1}\right)$$

Watch out for (3)

• x + a means substitute −a
  → Remainder on division by a linear factor
• The remainder is not a number
  → Remainder on division by a quadratic
• Even n and a plus sign
  → Divisibility of xⁿ ± aⁿ

Formulas (2)

• Testing for a factor · Factor theorem
  $$(x - a) \mid f(x) \iff f(a) = 0$$
• Finding unknown coefficients · Quadratic factor
  $$(x - \alpha)(x - \beta) \mid f(x) \iff f(\alpha) = f(\beta) = 0$$

Watch out for (2)

• A factor of one is not a factor of both
  → Testing for a factor
• Factorise the quadratic factor first
  → Finding unknown coefficients

Formulas (3)

• Cubics: find a root, then divide · Rational root test (monic)
  $$x^3 + bx^2 + cx + d:\ \text{integer roots divide } d$$
• Pair the brackets and substitute · Pairing four brackets
  $$x(x + 3)\cdot(x + 1)(x + 2) = t(t + 2), \quad t = x^2 + 3x$$
• Factorising with identities · A useful quartic
  $$a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)$$

Watch out for (3)

• Check the sign of each root
  → Cubics: find a root, then divide
• Pair by equal sums
  → Pair the brackets and substitute
• Plus or minus in the linear factor
  → Factorising with identities

Formulas (3)

• Reading off the HCF and LCM · HCF and LCM
  $$\text{HCF}: \text{common factors, lowest power}; \quad \text{LCM}: \text{all factors, highest power}$$
• HCF × LCM = product · Product rule
  $$p(x)\,q(x) = \text{HCF}(p, q)\times\text{LCM}(p, q)$$
• Unknowns from a given HCF · Common root
  $$k^2 + ak + b = 0,\ k^2 + ck + d = 0 \;\Rightarrow\; k = \dfrac{d - b}{a - c}$$

Watch out for (3)

• Lowest power for the HCF
  → Reading off the HCF and LCM
• Only for two polynomials
  → HCF × LCM = product
• x + k has root −k
  → Unknowns from a given HCF