Question:
What must be subtracted from the polynomial $f(x)=x^{4}+2 x^{3}-13 x^{2}-12 x+21$ so that the resulting polynomial is exactly divisible by $x^{2}-4 x+3 ?$
Solution:
We know that Dividend $=$ Quotient $\times$ Divisor $+$ Remainder .
Dividend $-$ Remainder $=$ Quotient $\times$ Divisor .
Clearly, Right hand side of the above result is divisible by the divisor.
Therefore, left hand side is also divisible by the divisor.
Thus, if we subtract remainder from the dividend, then it will be exactly divisible by the divisor.
Dividing $x^{4}+2 x^{3}-13 x^{2}-12 x+21$ by $x^{2}-4 x+3$
Therefore, quotient $=x^{2}+6 x+8$ and remainder $=(2 x-3)$
Thus, if we subtract the remainder $2 x-3$ from $x^{4}+2 x^{3}-13 x^{2}-12 x+21$, it will be divisible by $x^{2}-4 x+3$