What must be added to $f (x) = 4 x^{4} + 2 x^{3} - 2 x^{2} + x - 1$ so that the resulting polynomial is divisible by $g (x) = x^{2} + 2 x - 3$
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Solution

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

By division algorithm, we have

$f (x) = g (x) \times q (x) + r (x)$

$\Rightarrow f (x) - r (x) = g (x) \times q (x)$

$\Rightarrow f (x) + (- r (x))$ = g(x) \times q(x)\)

Clearly, RHS is divisible by $g (x)$ . Therefore, LHS is also divisible by $g (x)$ . Thus, if we add $- r (x)$ to $f (x)$ , then the resulting polynomial is divisible by $g (x)$ . Let us now find the remainder when $f (x)$ is divided by $g (x)$ .

$∴ r (x) = - 61 x + 65$

Hence, we should add $(- r (x) = 61 x - 65$ to $f (x))$ so that the resulting polynomial is divisible by $g (x)$ .

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what must be added to the polynomial $f (x) = 2 x^{3} - 2 x^{2} + x - 1$ so that the resulting polynomial is exactly divisible by $x^{2} + 2 x - 3$ ?

The expression that should be added to the polynomial f(x) = x⁴ + 2x³ – 2x² + x + 1, so that it should be exactly divisible by (x² + 2x – 3) is

x^{4} + 2 x^{3} - 2 x^{2} + x - 1

x^{2} + x - 2

p (x) = 2 x^{3} + 9 x^{2} - 5 x - 15

2 x + 3

4 x^{4} - 2 x^{3} - 6 x^{2} + x - 5

2 x^{2} + x - 1

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