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Theorem
If a polynomial p(x) of degree greater than or equal to one is divided by a linear polynomial (x - a) then the remainder is p(a), where a is any real number.
Explanation:
If a polynomial p(x) is divided by (x-a), then p(a) is the remainder.
The theorem states that, any polynomial p(x) when divided by an expression of the form (x-a) leaves a remainder p(a).
Example:
1. Find the remainder when x^{2} + 3x + 2 is divided by x + 2.
Given:
The polynomial p(x) = x^{2} + 3x + 2.
To find:
The remainder when p(x) = x^{2} + 3x + 2 is divided by x + 2.
Theorem used:
If a polynomial p(x) is divided by (x-a), then p(a) is the remainder.
Solution:
Step 1: Find the zero of the polynomial x = a.
Equate x + 2 to zero and solve for x.
x + 2 = 0
x = -2
Step 2: Find the remainder p(-2).
Substitute x = -2 in p(x).
p(-2) = (-2)^{2} + 3(-2) + 2
= 4 - 6 + 2
= 0
2. Find the remainder when x^{2} + 3x - 2 is divided by x + 1.
Given:
The polynomial p(x) = x^{2} + 3x - 2.
To find:
The remainder when p(x) = x^{2} + 3x - 2 is divided by x + 1.
Theorem used:
If a polynomial p(x) is divided by (x-a), then p(a) is the remainder.
Solution:
Step 1: Find the zero of the polynomial x = a.
Equate x + 1 to zero and solve for x.
x + 1 = 0
x = -1
Step 2: Find the remainder p(-1).
Substitute x = -1 in p(x).
p(-1) = (-1)^{2} + 3(-1) - 2
= 1 - 3 - 2
= -4
Important!
In Example (1), x + 2 is the factor of the polynomial x^{2} + 3x + 2 as it satisfies the equation p(x) = 0.
In Example (2), x + 1 is not the factor of the polynomial x^{2} + 3x - 2 as it does not satisfies the equation p(x) = 0.
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