Question

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Answer 1

EXPLANATION.

Without actual division.

2x⁴ - 5x³ + 2x² - x + 2 is exactly divisible by x² - 3x + 2.

As we know that,

⇒ x² - 3x + 2 = 0.

Factorizes the equation into middle term splits, we get.

⇒ x² - 2x - x + 2 = 0.

⇒ x(x - 2) - 1(x - 2) = 0.

⇒ (x - 1)(x - 2) = 0.

⇒ x = 1  and  x = 2.

Put the value of x = 1 in the equation, we get.

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

⇒ p(1) = 2 - 5 + 2 - 1 + 2.

⇒ p(1) = 6 - 6.

⇒ p(1) = 0.

Put the value of x = 2 in the equation, we get.

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

⇒ p(2) = 32 - 40 + 8 - 2 + 2.

⇒ p(2) = 42 - 42.

⇒ p(2)  0.

Hence, it is exactly divisible by x² - 3x + 2.