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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