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(a + b)^2 Expansion and Visualisation

If the polyno...

Question

If the polynomials $(2 x^{3} + a x^{2} + 3 x - 5)$ and $(x^{3} + x^{2} - 4 x - a)$ leave the same reminds when divided by (x - 1), find the value of a.

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Solution

Case 2 lets divide
$2 x^{2} + (a + 2) x + (a + 5) x - 1) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 2 x^{3} + a x^{2} + 3 x - 5 (2 x^{3} - 2 x^{2}0 + x^{2} (2 + a) + 3 x x^{2} (a + 2) - (a + 2) x+ x (3 + a + 2) - 5 = x (a + 5) - 5 x (a + 5) - (a + 5)0 + a + 5 - 5 = a$ $\to$ Remainder $1$
Case 2 $x^{2} + 2 x - 2 x - 1) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{3} + x^{2} - 4 x - a (x^{3} - x^{2}02 x^{2} - 4 x 2 x^{2} - 2 x0 - 2 x - a - 2 x + 20 - a - 2$ $\to$ Remainder $2$
Since Remainders are equal
$a = - a - 2$
$2 a = - 2$
$a = - 1$

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