Polynomials • Topic 3 of 4

Division of Polynomials and Remainder Theorem

What is Polynomial Division?

Polynomial division is similar to long division with numbers. When dividing a polynomial by a linear divisor (x - a), we can use two methods:

Division Algorithm for Polynomials:

When dividing polynomial P(x) by divisor d(x), we get:

P(x) = d(x) × Q(x) + R(x)

Where Q(x) is the quotient and R(x) is the remainder. The degree of R(x) is less than the degree of d(x).

What is a Zero of a Polynomial?

A zero (or root) of a polynomial P(x) is a value of x for which P(x) = 0.

If P(a) = 0, then a is a zero of the polynomial
Geometrically, a zero means the graph of the polynomial crosses the x-axis at x = a

Remainder Theorem:

When a polynomial P(x) is divided by (x - a), the remainder is P(a).

This means: P(x) = (x - a) × Q(x) + P(a)

Factor Theorem (special case of Remainder Theorem):

If P(a) = 0, then (x - a) is a factor of P(x).

Solved Examples

Worked Example

Solve a standard problem on Division of Polynomials and Remainder Theorem.

Solution

Apply the formula/method shown in the concept section above.

Understand the definition and properties of Division of Polynomials and Remainder Theorem.
Study the worked examples and practice similar problems.
Always verify your answer using the original conditions.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The remainder when $p(x)$ is divided by $(x-a)$ is:

Explanation: Remainder theorem.

Q2.$(x-a)$ is a factor of $p(x)$ iff:

Explanation: Factor theorem.

Q3.The remainder of $x^2+1$ divided by $(x-1)$ is:

Explanation: $p(1)=2$.

Q4.Division algorithm: dividend $=$ divisor $\times$ quotient $+$:

Explanation: $+$ remainder.