Polynomials: Remainder Theorem, Factor Theorem, Roots and Coefficients and Algebraic Identities for SSC CGL Algebra

Polynomials are the heart of SSC algebra, with questions on the remainder theorem, the factor theorem, the roots and the identities. This guide on polynomials, the remainder theorem and the factor theorem for SSC covers the key definitions, the theorems, the relations between the roots and the coefficients and the important identity-based question types, with solved examples, practice questions and FAQs.

Basics

Remainder Theorem

If a polynomial p(x) is divided by (x − a), the remainder is p(a). If it is divided by (ax − b), the remainder is p(b/a).

Factor Theorem

(x − a) is a factor of p(x) if and only if p(a) = 0. This is used to factorise cubic and higher polynomials: find one root by trial (generally ±1, ±2, ±3) and divide.

Roots and Coefficients

• Nature of the roots of a quadratic: discriminant D = b² − 4ac; D > 0 gives two real distinct roots, D = 0 gives equal roots and D < 0 gives no real roots.
• If one root of a quadratic with rational coefficients is p + √q, the other root is p − √q.
• If the roots are reciprocal of each other, then a = c; if the roots are equal in magnitude and opposite in sign, then b = 0.

Important Identities

Questions on x + 1/x

Solved Examples

Example 1. Find the remainder when x³ + 2x² − 5 is divided by (x − 2).
p(2) = 8 + 8 − 5 = 11.

Example 2. Show that (x − 1) is a factor of x³ − 6x² + 11x − 6, and find the other factors.
p(1) = 1 − 6 + 11 − 6 = 0, so it is a factor. The polynomial = (x − 1)(x − 2)(x − 3).

Example 3. If x + 1/x = 3, find x² + 1/x² and x³ + 1/x³.
x² + 1/x² = 9 − 2 = 7; x³ + 1/x³ = 27 − 9 = 18.

Example 4. Find a³ + b³ + c³ if a = 2, b = 3 and c = −5.
a + b + c = 0, so the value = 3abc = 3 × 2 × 3 × (−5) = −90.

Example 5. For 2x² − 7x + 3 = 0, find the sum and the product of the roots, and the roots.
Sum = 7/2; product = 3/2; the roots are 3 and 1/2.

Example 6. Find the value of k for which (x + 2) is a factor of x³ + 3x² + kx + 10.
p(−2) = −8 + 12 − 2k + 10 = 0, so 14 − 2k = 0 and k = 7.

Quick Tips

• Use p(a) to find the remainder by division by (x − a).
• Try the roots ±1, ±2 and ±3 for the factorisation of cubics.
• Memorise the x + 1/x results.
• Use the a + b + c = 0 shortcut for the sum of cubes.

Quick Revision Points

• Remainder theorem: remainder = p(a).
• Factor theorem: p(a) = 0 means (x − a) is a factor.
• Sum of roots of a quadratic = −b/a; product = c/a.
• If a + b + c = 0, then a³ + b³ + c³ = 3abc.
• x² + 1/x² = (x + 1/x)² − 2.

Practice Questions With Answers

1. Find the remainder when x² + 3x + 5 is divided by (x + 1).
   Answer: 3
2. If x + 1/x = 4, find x² + 1/x².
   Answer: 14
3. If x + 1/x = 2, find x³ + 1/x³.
   Answer: 2
4. What is the sum of the roots of 3x² − 9x + 2 = 0?
   Answer: 3
5. What is the product of the roots of x² − 5x + 6 = 0?
   Answer: 6
6. Find k if (x − 3) is a factor of x² − kx + 6.
   Answer: 5
7. If a + b + c = 0, find (a³ + b³ + c³)/abc.
   Answer: 3
8. What is the degree of the polynomial 5x⁴ − 2x + 7?
   Answer: 4
9. If x − 1/x = 3, find x² + 1/x².
   Answer: 11
10. What is the discriminant of x² − 4x + 4?
    Answer: 0

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Mathematics notes page, read the previous topic Quadrilaterals and Polygons and the next topic Inequalities and Modulus.

Frequently Asked Questions

What is a polynomial?

An algebraic expression with the variable raised to whole-number powers.

What is the remainder theorem?

If p(x) is divided by (x − a), the remainder is p(a).

What is the factor theorem?

(x − a) is a factor of p(x) when p(a) = 0.

What is the sum of the roots of a quadratic?

−b/a.

What is the product of the roots of a quadratic?

c/a.

What is the discriminant?

b² − 4ac; it tells the nature of the roots.

What is the value of x² + 1/x² if x + 1/x = k?

k² − 2.

What is the identity for a³ + b³ + c³ − 3abc?

(a + b + c)(a² + b² + c² − ab − bc − ca).

What happens to a³ + b³ + c³ if a + b + c = 0?

It equals 3abc.

Is this topic asked in SSC?

Yes. Algebra questions on identities and polynomials are asked every year.

What is a zero of a polynomial?

A value of the variable for which the polynomial is zero.

How many roots can a cubic have?

At most three.