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Standard X

Mathematics

Zeroes of a Polynomial

If fx=x4-2x...

Question

If $f (x) = x^{4} - 2 x^{3} - a x + b$ is a polynomial such that when it is divided by $x - 1$ and $x + 1$ , the remainders are respectively $45$ and $19$ . Determine the remainder when $f (x)$ is divided by $(x - 2)$ .

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Solution

When $f (x)$ is divided by $x - 1$ the remainder is $b = 45$
When $f (x)$ is divided by $x + 1$ the remainder is $b + a + 3 = 19$
$∴ b = 45$ and $b + a + 3 = 19$
$\Rightarrow b + a = 19 - 3 = 16$
$\Rightarrow a = 16 - b = 16 - 45 = - 29$
$∴ a = - 29$ and $b = 45$
When $f (x)$ is divided by $x - 2$ the remainder is $103$

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If f(x)=x4−2x3−ax+b is a polynomial such that when it is divided by x−1 and x+1, the remainders are respectively 45 and 19. Determine the remainder when f(x) is divided by (x−2).

When f(x) is divided by x−1 the remainder is b=45When f(x) is divided by x+1 the remainder is b+a+3=19∴b=45 and b+a+3=19⇒b+a=19−3=16⇒a=16−b=16−45=−29∴a=−29 and b=45When f(x) is divided by x−2 the remainder is 103

If $f (x) = x^{4} - 2 x^{3} - a x + b$ is a polynomial such that when it is divided by $x - 1$ and $x + 1$ , the remainders are respectively $45$ and $19$ . Determine the remainder when $f (x)$ is divided by $(x - 2)$ .