Question

4 x cube + 2 X square + 3 X + 1 divide x minus 4

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Answer 1

Answer:

Divide the polynomial 2x

4

−4x

3

−3x−1 by (x−1) and verify the remainder with zero of the divisor.

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Let f(x)=2x

4

−4x

3

−3x−1

First see how many times 2x

4

is of x.

x

2x

4

=2x

3

Now multiply (x−1)(2x

3

)=2x

4

−2x

3

Then again see the first term of the remainder that is −2x

3

. Now do the same.

Here the quotient is 2x

3

−2x

2

−2x−5 and the remainder is −6.

Now, the zero of the polynomial (x−1) is 1.

Put x=1 in f(x),f(x)=2x

4

−4x

3

−3x−1

f(1)=2(1)

4

−4(1)

3

−3(1)−1

2(1)−4(1)−3(1)−1

=2−4−3−1

=−6

Is the remainder same as the value of the polynomial f(x) at zero of (x−1)?

From the above examples we shall now state the fact in the form of the following theorem.

It gives a remainder without actual division of a polynomial by a linear polynomial in one variable.

Given polynomial is f(x)=2x

4

−4x

3

−3x−1 and divided by (x−1)

Put x=1 in the given polynomial, we get

f(x)=2x

4

−4x

3

−3x−1

⇒f(x)=2(1)

4

−4(1)

3

−3(1)−1

⇒f(x)=2−4−3−1

⇒f(x)=−6

Then 2x

4

−4x

3

−3x−1 divided by (x−2) the reminder is −6.

Then polynomial 2x

4

−4x

3

−3x−1 divided by (x−2) the reminder is not zero.

Step-by-step explanation:

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