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Remainder Theorem

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While we study about the polynomials, we come across the Remainder Theorem of the polynomials. According to the Remainder Theorem we say that if we have a polynomial say f(x) of degree n, where n > = 1and “a” is any of the real number. Now we say that f(x) is divided by ( x – a ), then we say that we get the remainder as any polynomial f(a). To prove this we proceed as follows: When we divide the polynomial f(x) by (x – a) the quotient we get I g(x) and r(x) is the remainder for the division so performed. So clearly we have degree r(x) < degree ( x – a) So we say that the degree of r(x) < 1 as we know that the degree of x – a = 1 Thus we say that the degree of r(x) = 0
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Remainder Theorem
Remainder Theorem
While we study about the polynomials, we come across the Remainder Theorem of the
polynomials. According to the Remainder Theorem we say that if we have a polynomial say
f(x) of degree n, where n > = 1and "a" is any of the real number.
Now we say that f(x) is divided by ( x - a ), then we say that we get the remainder as any
polynomial f(a).
To prove this we proceed as fol ows:
When we divide the polynomial f(x) by (x - a) the quotient we get I g(x) and r(x) is the
remainder for the division so performed.
So clearly we have degree r(x) < degree ( x - a)
So we say that the degree of r(x) < 1 as we know that the degree of x - a = 1
Thus we say that the degree of r(x) = 0
Know More About :- Linear Approximation

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Page No. :- 1/4

So we say that r(x) is a constant value and so it is equal to r ( let us assume)
Now we say that that when f(x) is divided by ( x - a) , then we clearly have g(x) as the
quotient and r as the remainder of this division process.
So we can clearly say that
F(x) = ( x - a ) . g(x) + r
On putting the value of x = a in the given equation, we get
F(a) = ( a - a ) . g(x) + r
Or F(a) = r
Thus we come to the conclusion that whenever we divide f(x) by ( x - a ), then we have the
remainder as f (a).
This theorem of polynomial is cal ed the remainder theorem of the polynomials and we use
this remainder theorem in finding the value of the remainder.
if we know the dividend and the divisor of the division. In case we get the remainder as zero,
we come to the conclusion that the divisor is the perfect factor of the dividend.

By remainder theorem we say that if we are given any polynomial f(x) = x^4 + 2 x^3 - 3.x^2 +
x - 1, then we say that if the polynomial is divided by another polynomial (x - 2), and we
need to find the remainder after the, then we will first find the zero of the polynomial (x - 2)
and the value which we will get for x, wil be placed into the given polynomial, then the result.
So obtained wil be the value of the remainder, which we wil get by dividing the polynomial
f(x) by the polynomial (x - 2 )

Math.Tutorvista.com
Page No. :- 2/4

Here by putting (x - 2 ) = 0, we get x = 2. So we say that at x = 2, this polynomial will be
equal to zero.
Now by the remainder Theorem we need to find the remainder of the division. So for this we
wil put this value of x = 2 in the polynomial f(x) = x^4 + 2 x^3 - 3.x^2 + x - 1
We get the remainder as fol ows : 2^4 + 2 * 2^3 - 3 * 2^2 + 2 - 1
= 16 + 16 - 12 + 2 - 1
= 34 - 13
= 21 Ans

Math.Tutorvista.com
Page No. :- 4/4

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Remainder Theorem

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