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When x^(5)-5x^(4)+9x^(3)-6x^(2)-16x+13 i...

When `x^(5)-5x^(4)+9x^(3)-6x^(2)-16x+13` is divided by `x^(2)-3x+a`, then quotient and remainders are `x^(3)-2x^(2)+x+1` and -15x+11 respectively. Find the value of a.

Text Solution

Verified by Experts

We know that
Dividend = Divisor `xx` Quotient + Remainder
`therefore x^(5)-5x^(4)+9x^(3)-6x^(2)-16x+3=(x^(2)-3x+a)(x^(3)-2x^(2)+x+1)+(-15x+11)`
`implies x^(5)-5x^(4)+9x^(3)-6x^(2)-16x+13=x^(5)-5x^(4)+(7+a(x^(3)-(2+2a)x^(2)+(-3+a-15)x+(a+11)`
Comparing the coefficent of `x^(3)` on both sides we get
`7+a=9 implies a=2`
(You can also compare the coefficient of any term involving 'a', everytime you will get the same 'a').
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