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Without actual division, prove that 2x⁴ - 5x³ + 2x² - x + 2 is divisible by x² - 3x + 2.

Solution:

Given, the polynomial is 2x⁴ - 5x³ + 2x² - x + 2

We have to prove that the polynomial is divisible by x² - 3x + 2 without actual division.

Given, the factor is x² - 3x + 2

On factoring x² - 3x + 2 by splitting the middle term,

x² - 3x + 2 = x² - 2x - x + 2

= x(x - 2) - 1(x - 2)

= (x - 1)(x - 2)

Therefore, the factors are (x - 1) and (x - 2)

Let p(x) = 2x⁴ - 5x³ + 2x² - x + 2

Let q(x) = x - 1

We know q(x) = 0

x - 1 = 0

x = 1

Put x = 1 in p(x),

p(1) = 2(1)⁴ - 5(1)³ + 2(1)² - (1) + 2

= 2 - 5 + 2 - 1 + 2

= 2 + 2 + 2 - 5 - 1

= 6 - 6

p(1) = 0

Let r(x) = x - 2

We know r(x) = 0

x - 2 = 0

x = 2

Put x = 2 in p(x),

p(2) = 2(2)⁴ - 5(2)³ + 2(2)² - (2) + 2

= 2(16) - 5(8) + 2(4) - 2 + 2

= 32 - 40 + 8

= 32 + 8 - 40

= 40 - 40

p(2) = 0

Therefore, p(1) = p(2) = 0

✦ Try This: Show that 2x+1 is a factor of polynomial 2x(cube) - 11x(square) - 4x + 1.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 2

NCERT Exemplar Class 9 Maths Exercise 2.4 Problem 4

Without actual division, prove that 2x⁴ - 5x³ + 2x² - x + 2 is divisible by x² - 3x + 2

Summary:

Without actual division, it is proven that 2x⁴ - 5x³ + 2x² - x + 2 is divisible by x² - 3x + 2 since p(1) = p(2) = 0

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