If the polynomials $m x^{3} - 2 x^{2} + 25 x - 26$ and $2 x^{3} - m x + 9$ leave the same remainder when they are divided by $(x - 2)$ , find the value of m. Also find the remainder.

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Solution

Let, $f (x) = m x^{3} - 2 x^{2} + 25 x - 26$
$f (2) = m (2)^{3} - 2 (2)^{2} + 25 (2) - 26$
$= 8 m - 8 + 50 - 26$
$= 8 m + 16$

Let, $p (x) = 2 x^{3} - m x + 9$
$p (2) = 2 (2)^{3} - 2 m + 9$
$= 16 + 9 - 2 m = 25 - 2 m$

Given: $f (x)$ and $p (x)$ leave the same remainder, when they are divided by $(x - 2)$ ,
$∴ f (2) = p (2)$
$⟹ 8 m + 16 = 25 - 2 m$
$⟹ 8 m + 2 m = 25 - 16$
$⟹ 10 m = 9$
$⟹ m = \frac{9}{10}$
Now,
Remainder $= f (2) = 8 m + 16$
$= 8 \times \frac{9}{10} + 16$
$= 7.2 + 16 = 23.2$

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If the polynomials mx3−2x2+25x−26 and 2x3−mx+9 leave the same remainder when they are divided by (x−2), find the value of m. Also find the remainder.

If the polynomials $m x^{3} - 2 x^{2} + 25 x - 26$ and $2 x^{3} - m x + 9$ leave the same remainder when they are divided by $(x - 2)$ , find the value of m. Also find the remainder.