Polynomials: SSC CGL Quantitative Aptitude

Formulas (4)

• Value of a polynomial at given numbers · Same-degree sign rule
  $$P(-x, -y) = (-1)^d \, P(x, y)$$
• Remainder theorem and factor theorem · Remainder and factor theorems
  $$p(x) \div (x - a) \Rightarrow \text{remainder} = p(a) \qquad (x - a) \text{ is a factor} \Leftrightarrow p(a) = 0$$
• Sum and product of the roots from the coefficients · Roots and coefficients
  $$ax^2 + bx + c: \ \alpha + \beta = -\frac{b}{a},\ \alpha\beta = \frac{c}{a} \qquad ax^3 + bx^2 + cx + d: \ p + q + r = -\frac{b}{a},\ pq + qr + rp = \frac{c}{a},\ pqr = -\frac{d}{a}$$
• A product of brackets that collapses by the difference of squares · Difference-of-squares chain
  $$(a - b)(a + b) = a^2 - b^2 \qquad (1 - p)(1 + p)(1 + p^2)(1 + p^4) = 1 - p^8$$

Watch out for (9)

• Losing the sign of an odd power
  → Value of a polynomial at given numbers
• Using the sign rule when the degrees differ
  → Value of a polynomial at given numbers
• Putting x = a for the divisor x + a
  → Remainder theorem and factor theorem
• Subtracting the wrong number to make a polynomial divisible
  → Remainder theorem and factor theorem
• Assuming x + 1 divides every xⁿ + 1
  → Remainder theorem and factor theorem
• Reading the sum of roots with the coefficient's own sign
  → Sum and product of the roots from the coefficients
• Forgetting to divide by the leading coefficient
  → Sum and product of the roots from the coefficients
• Trying to collapse a chain with no minus bracket
  → A product of brackets that collapses by the difference of squares
• Adding the exponents instead of doubling them
  → A product of brackets that collapses by the difference of squares