Remainder Theorem
The Remainder Theorem is a key concept in algebra that provides a simple and efficient way to find the remainder when a polynomial is divided by a linear divisor. It plays an important role in polynomial division, which is a fundamental operation in algebraic manipulations and problem-solving. The theorem is particularly useful in simplifying polynomial expressions and solving equations where division by a polynomial is involved.
What is the Remainder Theorem?
The Remainder Theorem states that when a polynomial π(π₯) is divided by a linear divisor of the form (π₯ β π), the remainder of the division is simply π(π). In other words, the remainder when a polynomial is divided by (π₯ β π) is the value of the polynomial evaluated at π₯ = π.
Mathematically, the Remainder Theorem can be expressed as:
If π(π₯) is divided by (π₯ β π), then the remainder is π(π).
This theorem is incredibly useful because it gives a quick way to find the remainder of polynomial division without needing to perform the entire long division process.
Statement of the Remainder Theorem
Given a polynomial π(π₯) of degree π, when π(π₯) is divided by (π₯ β π), the remainder π is given by:
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π(π₯) = (π₯ β π) β π(π₯) + π where:
β’ π(π₯) is the polynomial being divided,
β’ (π₯ β π) is the divisor, β’ π(π₯) is the quotient,
β’ π is the remainder.
According to the Remainder Theorem, the remainder π is simply π(π), so we have:
π(π) = π
Thus, to find the remainder when dividing π(π₯) by (π₯ β π), you just need to evaluate π(π).
Example of Applying the Remainder Theorem
Let's apply the Remainder Theorem to a specific example to see how it works in practice.
Example:
Divide π(π₯) = 2π₯Β³ β 3π₯Β² + 4π₯ β 5 by π₯ β 2.
Solution: To find the remainder, we evaluate π(π₯) at π₯ = 2:
π(2) = 2(2)Β³ β 3(2)Β² + 4(2) β 5
First, calculate the powers of 2:
π(2) = 2(8) β 3(4) + 4(2) β 5
π(2) = 16 β 12 + 8 β 5
π(2) = 7
Thus, the remainder when dividing 2π₯Β³ β 3π₯Β² + 4π₯ β 5 by π₯ β 2 is 7.
Proof of the Remainder Theorem:
Let's briefly prove the Remainder Theorem using polynomial division.
Given a polynomial π(π₯), we divide it by (π₯ β π). According to the division algorithm for polynomials, we can write:
π(π₯) = (π₯ β π) β π(π₯) + π
where π(π₯) is the quotient and π is the remainder. Since we are dividing by a linear polynomial (π₯ β π), the degree of the remainder π must be less than the degree of (π₯ β π), which is 1. Therefore, π must be a constant.
Now, substitute π₯ = π into the equation:
π(π) = (π β π) β π(π) + π π(π) = 0
β π(π) + π π(π) = π
Thus, the remainder when dividing π(π₯) by (π₯ β π) is π(π), proving the Remainder Theorem.
Applications of the Remainder Theorem:
1. Evaluating Polynomial Remainders Quickly: As seen in the example above, the Remainder Theorem allows us to quickly find the remainder of a polynomial division by simply evaluating the polynomial at a specific value of π₯, rather than performing full polynomial long division.
2. Solving Polynomial Equations: The Remainder Theorem can help solve polynomial equations. For instance, if a polynomial π(π₯) is divided by (π₯ β π) and the remainder is zero, then π₯ = π is a root of the polynomial.
3. Finding Factors of Polynomials: If π(π) = 0, it implies that π₯ β π is a factor of the
polynomial π(π₯). This can be useful in factoring polynomials or finding roots of
equations.
4. Testing for Divisibility: The Remainder Theorem is also helpful in testing
whether one polynomial is divisible by another. If π(π) = 0, then π(π₯) is divisible
by (π₯ β π), and π₯ = π is a root of the polynomial.
The Factor Theorem:
The Factor Theorem is a direct consequence of the Remainder Theorem. It states that if π(π) = 0, then π₯ β π is a factor of the polynomial π(π₯). In other words, if the remainder is zero, the divisor is a factor of the polynomial. If π(π) = 0, then (π₯ β π) is a factor of π(π₯).
Example of the Factor Theorem Suppose we have the polynomial π(π₯) = π₯Β² β 5π₯ + 6 and want to check if π₯ β 2 is a factor of π(π₯). We apply the Remainder Theorem by evaluating π(π₯) at π₯ = 2:
π(2) = (2)Β² β 5(2) + 6
π(2) = 4 β 10 + 6 = 0
Since π(2) = 0, according to the Factor Theorem, π₯ β 2 is a factor of π(π₯).
FAQs: Remainder Theorem
What is the difference between the Remainder Theorem and the Factor Theorem?
The Remainder Theorem tells us that the remainder when dividing a polynomial by (x - a) is equal to f(a). The Factor Theorem is a special case of the Remainder Theorem which states that if f(a) = 0, then x - a is a factor of f(x).
Can the Remainder Theorem be used for division by any divisor?
The Remainder Theorem specifically applies to division by a linear divisor of the form (x - a). For higher-degree divisors, polynomial division must be used, and the Remainder Theorem does not directly apply.
How is the Remainder Theorem useful in solving polynomial equations?
The Remainder Theorem allows you to check quickly whether a particular value is a root of the polynomial. If f(a) = 0, then x = a is a root, which helps in factoring the polynomial or solving the equation.
Can we use the Remainder Theorem with higher-degree polynomials?
Yes, the Remainder Theorem can be used for any polynomial. It works for polynomials of any degree, but it is particularly useful when the divisor is a linear polynomial.