Remainder Theorem formula proof and step by step solved examples
The polynomial remainder theorem states that when any polynomial p(x) with a degree of one or a greater number is divided by (x - a), a linear polynomial where a is any real number, you obtain p(a) as a remainder.
When it comes to the Euclidean division, the division of real numbers is fairly simple. You take a number, say 24, divide it by 5. You get a quotient of 4 and a remainder of 4. Thus, you can conclude that 24 = (5 x 4) + 4. If you divide the same 24 by 4, you get a quotient of 6 and a remainder of 0. So both 4 and 6 are factors of 24, and 24 is a multiple of both 4 and 6.
But when it comes to the Euclidean division of polynomials, things can get long and complicated. The remainder theorem and Factor theory are concepts that make this easier for you.
Understanding Remainder Theorem
Polynomial remainder theorem, otherwise known as little Bezout’s theorem gives us a method of identifying the remainder of a polynomial divided by a linear equation. If we divide a polynomial p(x) with a linear equation (x-a), the resulting remainder would be p(a). If the p(a) is 0, it means that the linear equation (x-a) is a factor of the polynomial p(x) (this is called factor theorem).
Let’s take a polynomial equation p(x) = x² + 6x - 3, when you divide with a linear polynomial x-3, the remainder should be p(3).
p(3) = (3)² + 6(3) - 3
= 9 + 18 - 3
= 24
The remainder is now 24.
Long Divison Verification
Let’s verify this now with the traditional long division method:
x+9
x-3 \[\sqrt{x^{2} + 6x - 3}\]
- x² - 3x
9x - 3
- 9x - 27
= 24
So here, we have our p(x) = x² + 6x - 3 divided by x - 3 in the long division method giving us a quotient of x+9 and a remainder 24.
Thus we can verify that p(x) = x² + 6x - 3 divided by (x - 3) will give us a reminder p(3).
You can verify this with other polynomials too. Before you divide a polynomial with a non-zero linear equation, make sure that:
The terms of a polynomial are arranged in the descending order of their degrees.
You divide the first term of the polynomial dividend with your divisor’s first term to obtain your first quotient.
The resulting terms after the subtraction act as your next dividend while the divisor remains the same.
Continue the process until your dividend has a lesser degree than that of your divisor.
Factor Theorem
As we hinted earlier, the factor theorem is basically the inverse of the polynomial remainder theorem. As we discussed earlier when you divide 24 with 4 you get a remainder of 0, thus concluding 4 being a factor of 24. Similarly, if you divide a polynomial p(x) with a linear equation (x-a) and get the remainder as zero, it means that the linear equation x-a is a factor of the polynomial.
Remainder Theorem Formula and Proof
So from our understandings so far, we can identify that a remainder theorem equation would be:
p(x) = (x-a) * q(x) + r(x)
Where r(x) equals p(a).
Now let us prove this. Since r(x) is a constant, it can just be r. Now for our p(a)
p(a) = (a-a) * q(a) + r
= 0 * q(a) + r
= r
Thus we get the result p(a) = r, the remainder.
Remainder Theorem Examples
Let us now take a look at a couple of remainder theorem examples with answers.
Example 1:
What would be the remainder when you divide x³+4x²-2x + 5 by x-5?
Solution:
p(x)= x³+4x²-2x+5
Divisor = x-5
p(5) = (5)³ + 4 (5)² - 2 (5) +5 = 125 + 100 - 10 + 5 = 220
Example 2:
What would be the remainder when you divide 3x²+15x-45 by x-15?
Solution:
p(x) = 3x²+15x-45
Divisor = x-15
p(15) = 3 (15)² + 15 (15) - 45 = 675 + 225 - 45 = 855
In this way, the remainder theorem has made it easy for us all to find the remainders of polynomial equations divided by linear equations without having to resort to the more complex long division method.
FAQs on Remainder Theorem in Polynomials Explained Clearly
1. What is the Remainder Theorem in polynomials?
The Remainder Theorem states that when a polynomial f(x) is divided by (x − a), the remainder is f(a).
- If you substitute x = a into the polynomial, the result equals the remainder.
- This theorem applies only when the divisor is linear and of the form (x − a).
- It simplifies polynomial division by avoiding long division.
2. How do you use the Remainder Theorem to find the remainder?
To find the remainder using the Remainder Theorem, substitute the value of a into f(x) where the divisor is (x − a).
- Step 1: Identify a from the divisor (x − a).
- Step 2: Compute f(a).
- Step 3: The result is the remainder.
3. What is the formula for the Remainder Theorem?
The formula for the Remainder Theorem is R = f(a) when a polynomial f(x) is divided by (x − a).
- R represents the remainder.
- a is the value from the divisor.
- This applies only to linear divisors of the form (x − a).
4. What is the difference between the Remainder Theorem and the Factor Theorem?
The Factor Theorem is a special case of the Remainder Theorem where the remainder equals zero.
- Remainder Theorem: Remainder = f(a).
- Factor Theorem: If f(a) = 0, then (x − a) is a factor.
5. Can you give an example of the Remainder Theorem?
Yes, the Remainder Theorem says the remainder equals f(a) when dividing by (x − a).
- Let f(x) = x³ − 4x + 1.
- Divide by (x + 2), so a = −2.
- Compute f(−2) = −8 + 8 + 1 = 1.
6. How do you check if a polynomial is a factor using the Remainder Theorem?
To check if (x − a) is a factor, calculate f(a) and see if it equals zero.
- If f(a) = 0, then (x − a) is a factor.
- If f(a) ≠ 0, it is not a factor.
7. Does the Remainder Theorem work for divisors other than (x − a)?
The Remainder Theorem works only for linear divisors of the form (x − a).
- It does not directly apply to quadratic or higher-degree divisors.
- For higher-degree divisors, use polynomial long division or synthetic division (if applicable).
8. What is synthetic division and how is it related to the Remainder Theorem?
Synthetic division is a shortcut method for dividing a polynomial by (x − a) and it directly uses the Remainder Theorem.
- You write the coefficients of the polynomial.
- Use the value a for division.
- The final number obtained is f(a), the remainder.
9. Why is the Remainder Theorem important in algebra?
The Remainder Theorem is important because it simplifies polynomial evaluation and helps find factors and roots efficiently.
- It avoids lengthy polynomial long division.
- It supports the Factor Theorem and solving polynomial equations.
- It is widely used in algebra and higher mathematics.
10. What are common mistakes when using the Remainder Theorem?
A common mistake in using the Remainder Theorem is choosing the wrong value of a from the divisor.
- If the divisor is (x + 3), then a = −3, not 3.
- Forgetting missing terms (like 0x²) in synthetic division.
- Applying the theorem to non-linear divisors.
