In this article, you will learn about the concept of the Remainder Theorem. In Maths, the Remainder Theorem is a way of addressing Euclidean’s division of polynomials. The other name of the Remainder Theorem is Bezout’s theorem, it approaches polynomials of Euclidean’s division. The remainder theorem is a formula that is used to find the remainder of a polynomial when a polynomial is divided by a linear polynomial.
When a certain number of things are divided into groups with an equal number of things in each group, the number of leftover things is known as the remainder. It is something that we called “remains” after the division. On this page, we will discuss the 10th Grade Math Remainder theorem definition, the formula of the remainder theorem, the proof of a Remainder theorem, how does remainder theorem works, Some example problems, and so on.
Read More: Polynomial Equation and its Roots
Remainder Theorem Definition
The remainder theorem definition states that when a polynomial of a(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = a(k). It enables us to calculate the remainder of the division of any polynomial by a linear polynomial, without actually carrying out the steps of the division algorithm.
In other words, the Remainder Theorem Definition states that when a polynomial is p(a) is divided by another binomial ( a – x ), then the remainder of the ending result is obtained by p (x). The resultant obtained is the value of the polynomial p(x) where x = a and this is possible only if p(a) = 0. In order to factorize the polynomials easily, we will apply the remainder theorem.
Factor Theorem: The Factor Theorem is generally applied to factoring and finding the roots of polynomial equations. It is the reverse form of the remainder theorem. The problems are solved based on the application of synthetic division and then checking for a zero remainder. When p(x) = 0 then y-x is a factor of the polynomial Or if we consider the other way, then When y-x is a factor of the polynomial then p(x) =0
Remainder Theorem Formula
The below is general formula for remainder theorem,
p(x) = (x-c)·q(x) + r(x).
- when p(x) is divided by (x-a) then the remainder will be p(a).
- When p(x) is divided by (ax+b) then the remainder will be p(-b/a).
Remainder Theorem Proof
The remainder Theorem functions on an actual case are, that a polynomial is comprehensively dividable, at least one time by its factor in order to get a smaller polynomial and ‘a’ remainder of zero. This acts as one of the simplest ways to determine whether the value ‘a’ is a root of the polynomial P(x). when we divide p(x) by x-a we can obtain
p(x) = (x-a)·q(x) + r(x),
As we know, Dividend = (Divisor × Quotient) + Remainder. If r(x) will be simply the constant of r (remember that when we divide by (x-a) the remainder is a constant). So, we obtain the following solution, which is
p(x) = (x-a)·q(x) + r
Now, observe what happens when we have x equal to a:
p(a) = (a-a)·q(a) + r
p(a) = (0)·q(a) + r
p(a) = r
Hence, it proved.
How Does Remainder Theorem Work?
Let us understand how the remainder theorem works, Consider a general case. Let a(x) be the dividend polynomial and b(x) be the linear divisor polynomial, and let q(x) be the quotient and r will be the constant reminder. Then, we have
a(x) = b(x). q(x) + r.
Next, let us denote the zero of the linear polynomial b(x) by k. This means that b(k) = 0. If we plug in ‘x’ as ‘k’ in the starred relation above, we have a(k) = b(k) q(k) + r.
observed that doing this is allowed since the starred relation will hold the truth for every value of x. In reality, it is a polynomial identity. Since b(k)=0 we are left with a(k)=r. In other words, the remainder is equal to the value of a(x) when x is equal to k, precisely what we stumbled upon! This is exactly what the remainder theorem is: When a polynomial a(x) is divided by a polynomial b(x) whose zero is x equal to k, the remainder is given by r=a(k).
Remainder Theorem Examples with Answers
Problem 1:
Find the remainder when x4 + x3 – 2x2 + x + 1 is divided by (x – 1)?
Solution:
As given in the question,
The Dividend Polynomial is p(x) = x4 + x3 – 2x2 + x + 1, and
The Divisor Polynomial is x – 1.
Now, we will find the value of the remainder.
The zero of the divisor polynomial is x – 1 = 0 or x = 1.
Substitute the value of ‘x’ in the dividend polynomial, we get
Therefore, p(1) = (1)4 + (1)3 – 2(1)2 + 1 + 1 = 1 + 1 – 2 + 1 + 1 = 2.
So, from the Remainder Theorem, the remainder will be 2.
Problem 2: Find the remainder when (x3 – ax2 + 6x – a) is divided by (x – a).
Solution:
Given that,
The Dividend Polynomial is p(x) = x3 – ax2 + 6x – a, and
The Divisor Polynomial is x – a.
Now, we need to find out the value of the remainder.
The zero of the divisor polynomial is x – a = 0 or x = a.
Replace the value of x in the dividend polynomial.
Therefore, p(a) = (a)3 – a(a)2 + 6a – a = a3 – a3 + 6a – a = 5a
Hence, by the Remainder Theorem, the remainder is 5a.
Problem 3: Find the root of the polynomial x2-3x-4?
Solution:
As given in the question,
The equation is p(x) = x2-3x-4
Let us consider that, the value of x be 4.
So that, f(4) = (4)2-3(4)-4
f(4) = 16-12-4 = 16-16 = 0.
Therefore, (x-4) must be a factor of the x2-3x-4.
Problem 4: If the polynomial expression is (x) = x4−2x3+3x2−ax+b such that when divided by x−1 and x+1, the remainder is respectively 5 and 19. Find the remainder value when f(x) is divided by (x−2).
Solution:
Given that,
The polynomial expression is f(x) = x4−2x3+3x2−ax+b.
If f(x) is divided by x-1 and x+1 then the remainders are 5 and 19 respectively.
Therefore f(1)=5 and f(−1)=19.
So, substitute the value of x as 1 in the given polynomial equation.
that is (1)4−2(1)3+3(1)2− a(1)+b = 5
Now, place the x value as −1, then it will be
(-1)4−2(−1)3+3(−1)2− a(−1)+b=19
Then the equations is
1−2+3−a+b=5
and 1+2+3+a+b=19
⇒2−a+b=5 and 6+a+b=19
⇒−a+b=3 and a+b=13
Adding these two equations, we get
(−a+b)+(a+b)=3+13
⇒2b=16 ⇒b=8
Now, Putting b=8 and −a+b=3, we get
−a+8=3 ⇒-a=−5 ⇒a=5
Putting the values of a and b in
f(x)=x4−2x3+3x2−5x+8
The remainder when f(x) is divided by (x-2) is equal to f(2).
So, Remainder =f(2)=(2)4−2(2)3+3(2)2−5(2)+8
= 16−16+12−10+8 =10
Hence, by dividing f(x) by x-2 we get the remainder is 10.
FAQ’s on Remainder Theorem
1. What is the Remainder Theorem formula?
The remainder theorem formula is: p(x) = (x-c)·q(x) + r(x). The basic formula to check the division is: Dividend = (Divisor × Quotient) + Remainder.
2. How do you use the remainder theorem?
The Remainder Theorem tells us that, in order to evaluate a polynomial p(x) at some number x = a, we can instead divide by the linear expression x − a. The remainder, r(a), gives the value of the polynomial at x = a.
3. What does the remainder theorem state?
The Remainder Theorem states that if polynomial ƒ(x) is divided by a linear binomial of the for (x – a) then the remainder will be f(a). The Factor Theorem states that if f(a) = 0 in this case the binomial (x – a) is the factor of polynomial f(x).
4. What is the importance of the remainder theorem and factor theorem?
The remainder theorem and factor theorem are convenient tools. They tell that we can find the factors of a polynomial without using long division, synthetic division, or other traditional methods of factoring. Using these theorems is somewhat of an error method and trial. The reverse form of the remainder theorem is the factor theorem.