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

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Solution

We divide the polynomial $x^{4} + 2 x^{3} - 2 x^{2} + x - 1$ by $x^{2} + 2 x - 3$ as shown in the above image:

From the above division, we conclude that the quotient is $x^{2} + 1$ and the remainder is $- x + 2$ .

Therefore, $- (- x + 2)$ that is $x - 2$ must be added to the given polynomial so that the resulting polynomial is exactly divisible by $x^{2} + 2 x - 3$ .

Hence, $(x - 2)$ must be added to the polynomial $x^{4} + 2 x^{3} - 2 x^{2} + x - 1$ so that the resulting polynomial is exactly divisible by $x^{2} + 2 x - 3$ .

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What must be added to the polynomial f(x)=x4+2x3−2x2+x−1 so that the resulting polynomial is exactly divisible by x2+2x−3 ?

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