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Using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the by actual division

Question:

Using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the by actual division

$f(x)=2 x^{4}-6 x^{3}+2 x^{2}-x+2, g(x)=x+2$

Solution:

Here, $f(x)=2 x^{4}-6 x^{3}+2 x^{2}-x+2$

g(x) = x + 2

from, the remainder theorem when f(x) is divided by g(x) = x - (-2) the remainder will be equal to f(-2)

Let, g(x) = 0

⟹ x + 2 = 0

⟹ x = - 2

Substitute the value of x in f(x)

$f(-2)=2(-2)^{4}-6(-2)^{3}+2(-2)^{2}-(-2)+2$

= (2 * 16) - (6 * (-8)) + (2 * 4) + 2 + 2

= 32 + 48 + 8 + 2 + 2

= 92

Therefore, the remainder is 92

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