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 and Polynomials
1. What is the Remainder Theorem as per the CBSE Class 9 syllabus?
The Remainder Theorem states that if a polynomial, let's say p(x), with a degree greater than or equal to one, is divided by a linear polynomial of the form (x - a), then the remainder of this division will be equal to p(a). In simpler terms, to find the remainder, you don't need to perform long division; you just need to substitute the zero of the divisor into the polynomial.
2. Can you explain how to apply the Remainder Theorem with an example?
Certainly. Let's find the remainder when the polynomial p(x) = x³ + 2x² - 4x + 1 is divided by (x - 1). According to the Remainder Theorem, the remainder is p(1). We substitute x = 1 into the polynomial:
p(1) = (1)³ + 2(1)² - 4(1) + 1
p(1) = 1 + 2 - 4 + 1
p(1) = 0
Therefore, the remainder is 0. This also means (x-1) is a factor of the polynomial.
3. What is the Factor Theorem and how is it connected to the Remainder Theorem?
The Factor Theorem is a direct consequence of the Remainder Theorem. It states that a linear polynomial (x - a) is a factor of a polynomial p(x) if and only if p(a) = 0. The connection is simple: the Factor Theorem is a special case where the remainder is zero. If applying the Remainder Theorem gives a result of 0, then the divisor is a factor of the polynomial.
4. What is the difference between a polynomial's 'degree' and its 'zeros'?
This is a common point of confusion. Here's the difference:
- The degree of a polynomial is the highest exponent of the variable in any term. It defines the type of polynomial (linear, quadratic, cubic) and its general shape.
- A zero of a polynomial is a specific value of the variable that makes the entire polynomial's value equal to zero. For example, in p(x) = x - 5, the degree is 1, and the zero is 5.
5. Are all algebraic expressions considered polynomials? Explain why or why not.
No, not all algebraic expressions are polynomials. For an expression to be classified as a polynomial, the exponents of its variables must be non-negative integers (0, 1, 2, 3,...). Expressions with variables in the denominator (like 1/x, which is x⁻¹) or under a radical sign (like √x, which is x¹/²) are not polynomials because they have negative or fractional exponents.
6. Why is the Remainder Theorem considered a more efficient method than long division?
The Remainder Theorem is a significant shortcut because it bypasses the entire, often lengthy, process of polynomial long division. Instead of dividing step-by-step to find both the quotient and the remainder, the theorem allows you to find the remainder directly through simple substitution and arithmetic. This saves considerable time and reduces the chances of calculation errors, making it a highly efficient tool, especially for objective questions in exams.
7. How does the Factor Theorem help in the factorisation of cubic polynomials?
The Factor Theorem is fundamental to breaking down cubic polynomials. The process involves these steps:
- First, use the 'trial and error' method to find one integer zero of the polynomial, say 'a'. You test the factors of the constant term.
- Once you find that p(a) = 0, you know from the Factor Theorem that (x - a) is a factor.
- Next, you divide the cubic polynomial by this known factor (x - a) using long division.
- The result will be a quadratic quotient, which can then be easily factorised further using standard methods like splitting the middle term.
8. What is the importance of identifying the degree of a polynomial?
Identifying the degree of a polynomial is crucial for several reasons. Firstly, it classifies the polynomial (e.g., degree 1 is linear, degree 2 is quadratic, degree 3 is cubic), which tells us about the general shape of its graph. Secondly, the Fundamental Theorem of Algebra states that the degree determines the maximum number of zeros (or roots) the polynomial can have. This is vital for solving polynomial equations and understanding their behaviour.
