The Remainder Theorem provides a straightforward method for finding the remainder when a polynomial is divided by a linear expression. Here's how to use it:
Understanding the Remainder Theorem
According to the Remainder Theorem, when a polynomial p(x) (where the degree is greater than or equal to 1) is divided by a linear polynomial x - a, the remainder is given by r = p(a). In simpler terms, to find the remainder, you simply substitute the value of 'a' into the polynomial p(x).
Steps to Find the Remainder
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Identify the Polynomial: Determine the polynomial, p(x), that you're working with.
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Identify the Linear Divisor: Determine the linear divisor in the form x - a. Pay close attention to the sign. If the divisor is x + a, it can be rewritten as x - (-a).
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Find 'a': Solve for 'a' from the linear divisor x - a = 0. This gives you x = a.
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Substitute and Evaluate: Substitute the value of 'a' into the polynomial p(x) and evaluate the expression. The result, p(a), is the remainder.
Examples
Example 1
Find the remainder when p(x) = x3 - 2x2 + x - 5 is divided by x - 3.
- p(x) = x3 - 2x2 + x - 5
- Divisor: x - 3
- a = 3
- p(3) = (3)3 - 2(3)2 + (3) - 5 = 27 - 18 + 3 - 5 = 7
Therefore, the remainder is 7.
Example 2
Find the remainder when p(x) = 2x4 + 5x3 - x2 + 3x + 1 is divided by x + 1.
- p(x) = 2x4 + 5x3 - x2 + 3x + 1
- Divisor: x + 1 (which is x - (-1))
- a = -1
- p(-1) = 2(-1)4 + 5(-1)3 - (-1)2 + 3(-1) + 1 = 2 - 5 - 1 - 3 + 1 = -6
Therefore, the remainder is -6.
In Summary
The Remainder Theorem offers a quick and efficient way to determine the remainder of polynomial division without performing long division or synthetic division. By simply substituting the value 'a' (derived from the linear divisor x - a) into the polynomial p(x), you can directly find the remainder p(a).
