Without actual division, prove that 2x^4 - 5x^3 + 2x^2 - x + 2 is… - Vidyadip

Question

Without actual division, prove that $2x^4 - 5x^3 + 2x^2 - x + 2$ is divisible $by x^2 - 3x + 2.$
[Hint: Factorise $x^2 - 3x + 2$]

✓

Answer

We have,
$x^2 - 3x + 2 = x^2 - x - 2x + 2$
$= x(x - 1) - 2(x - 1)$
$= (x - 1)(x - 2)$
Let $f(x) = 2x^4 - 5x^3 + 2x^2 - x + 2$
Now, $p(1) = 2(1)^4 - 5(1)^3 + 2(1)^2 - 1 + 2 = 2 - 5 + 2 - 1 + 2 = 0$
$p(1) = 0$
Therefore, $(x - 1)$ divides $p(x)$
And $p(2) = 2(2)^4 - 5(2)^3 + 2(2)^2 - 2 + 2$
$= 32 - 40 + 8 - 2 + 2 = 0$
$p(2) = 0$
Therefore, $(x - 2)$ divides $p(x).$
So, $(x - 1)(x - 2) = x^2 - 3x + 2$ divides $2x^2 - 5x^3 + 2x^2 - x + 2$

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