Question

the expression 2 x 4 - x3 - 3 x 2 + 5 x- 2 should be divisible by which expression so that the quotient is x2+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.

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