What is the Division Algorithm for Polynomials?

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

Home > Class 10 > Maths > Ch 2: Polynomials (Part 2)

Ch 2: Polynomials (Division Algorithm)

This page covers the remaining exercises from the traditional NCERT textbook: Division Algorithm for Polynomials (Old Ex 2.3).

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Old Exercise 2.3 Solutions

💡 Concept: Division Algorithm states that p(x) = g(x) × q(x) + r(x)

Q1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:

(i) p(x) = x³ - 3x² + 5x - 3, g(x) = x² - 2

Solution:

We divide x³ - 3x² + 5x - 3 by x² - 2

x - 3 _____________________ x² - 2 | x³ - 3x² + 5x - 3 x³ - 2x - (-) (+) ----------------------- - 3x² + 7x - 3 - 3x² + 6 (+) (-) ----------------------- 7x - 9

Quotient, q(x) = x - 3

Remainder, r(x) = 7x - 9

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Q3. Obtain all other zeroes of 3x⁴ + 6x³ - 2x² - 10x - 5, if two of its zeroes are √(5/3) and -√(5/3).

Most Important / 4 Marks

Solution:

Since √(5/3) and -√(5/3) are zeroes, then:
(x - √(5/3))(x + √(5/3)) is a factor of the given polynomial.

Using (a-b)(a+b) = a² - b²:
(x² - 5/3) = 0
Multiplying by 3, we get 3x² - 5 as a factor.

Now, we divide the given polynomial by (3x² - 5):

x² + 2x + 1 _________________________ 3x² - 5 | 3x⁴ + 6x³ - 2x² - 10x - 5 3x⁴ - 5x² - (-) (+) --------------------------- 6x³ + 3x² - 10x - 5 6x³ - 10x (-) (+) --------------------------- 3x² - 5 3x² - 5 (-) (+) --------------------------- 0

The quotient is x² + 2x + 1.

To find the remaining zeroes, we factorize the quotient:
x² + 2x + 1 = 0
(x + 1)² = 0
x = -1, -1

Therefore, all the zeroes are √(5/3), -√(5/3), -1, and -1.

Advanced PYQs (Long Division)

CBSE 2018

Q. On dividing x³ - 3x² + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and -2x + 4, respectively. Find g(x).

Solution:

Given:
Dividend, p(x) = x³ - 3x² + x + 2
Quotient, q(x) = x - 2
Remainder, r(x) = -2x + 4

By Division Algorithm:
p(x) = g(x) × q(x) + r(x)

Substituting the values:
x³ - 3x² + x + 2 = g(x) × (x - 2) + (-2x + 4)
x³ - 3x² + x + 2 + 2x - 4 = g(x) × (x - 2)
x³ - 3x² + 3x - 2 = g(x) × (x - 2)

Therefore, g(x) = (x³ - 3x² + 3x - 2) / (x - 2)

On dividing the numerator by (x - 2) using long division, we get the quotient:
x² - x + 1

Hence, g(x) = x² - x + 1.

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