Question

8. The expression 2x4 – x3 - 3x2 + 5x – 2 should be divisible by which expression so that the quotient is
7.
x + x - 1?​

Answer 1

Answer:

The division algorithm states that Dividend=Divisor×Quotient+Remainder that is f(x)=g(x)⋅q(x)+r(x)

Here, it is given that the dividend is f(x)=2x4−5x3+x2+3x−2, the divisor is 2x2−5x+3 and the remainder is −2x+1, therefore, by applying division algorithm we have:

2x4−5x3+x2+3x−2=(2x2−5x+3)g(x)+(−2x+1)

⇒2x4−5x3+x2+3x−2−(−2x+1)=(2x2−5x+3)g(x)

⇒2x4−5x3+x2+3x−2+2x−1=(2x2−5x+3)g(x)

⇒2x4−5x3+x2+5x−3=(2x2−5x+3)g(x)

⇒g(x)=2x2−5x+32x4−5x3+x2+5x−3 

Let us now divide 2x4−5x3+x2+5x−3 by 2x2−5x+3 as shown in the above image:

From the division, we observe that the quotient is x2−1 and the remainder is 0.

Hence, the quotient q(x)=x2−1.