Divide the polynomial 2x^4+5x^3-2x^2+2x-4 by (2x+1) and verify remainder using remainder theoram
Answers
Answer:
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