Remainder Theorem is a fundamental concept in algebra tha provides a shortcut for finding the remainder when a polynomial is divided by a linear expression.
It states that when you divide a polynomial \(f(x)\) by a linear expression \(x - c\), the remainder \(R\) is equal to \(f(c)\):
If you divide a polynomial \(f(x)\) by a linear expression \(x + c\), the remainder \(R\) is equal to \(f(-c)\):
Where \(f(x)\) is the polynomial to be divided, \(x-c\) or \(x + c\) is the linear divisor and \(c\) or \(-c\), respectively, is the point where the remainder is being evaluated.
Find the remainder of \(f(x) = 2x^3 - 5x^2 + 3x + 1 \) when it is divided by \( x-2 \).
First, we can identity which formula to use. Since the divisor is \(x-2\), we can use the formula \(R = f(c)\).
Next, we can solve for \(f(c)\). We can do this by substituting \(2\) for \(x\) in the main function:
\(R = f(2)\)
\(f(2) = 2(2)^3 - 5(2)^2 + 3(2) + 1 \)
\(f(2) = 2(8) - 5(4) + 6 + 1 \)
\(f(2) = 16 - 20 + 6 + 1 \)
\(f(2) = 3\)
Therefore, we can determine that the remainder when \(f(x)\) is divided by \(x-2\) is \(\boldsymbol{3}\).
\(f(x) = 2x^3 - 5x^2 + 3x + 1\) by \(x + 2\)
First, we can identity which formula to use. Since the divisor is \(x+2\), we can use the formula \(R = f(-c)\).
Next, we can solve for \(f(-c)\). We can do this by substituting \(-2\) for \(x\) in the main function:
\(R = f(-2)\)
\(f(-2) = 2(-2)^3 - 5(-2)^2 + 3(-2) + 1 \)
\(f(-2) = 2(-8) - 5(4) - 6 + 1\)
\(f(-2) = -16 - 20 - 6 + 1\)
\(f(-2) = -41\)
From this we can see the remainder when \(f(x)\) is divided by \(x + 2\) is \(\boldsymbol{-41}\)
\(f(x) = x^3 - 4x + 2\) by \(x - 2\)
First, we can identity which formula to use. Since the divisor is \(x-2\), we can use the formula \(R = f(c)\).
Next, we can solve for \(f(c)\). We can do this by substituting \(2\) for \(x\) in the main function:
\(R = f(2)\)
\(f(2) = (2)^3 -4(2) + 2\)
\(f(2) = (8) - 4(2) + 2\)
\(f(2) = 8 -8 + 2\)
\(f(2) = 2\)
From this we can see the remainder when \(f(x)\) is divided by \(x - 2\) is \(\boldsymbol{2}\)