What is the remainder when a³ – 4 is divided by a+2?
The Correct Answer and Explanation is :
To find the remainder when ( a^3 – 4 ) is divided by ( a + 2 ), we can use synthetic division or polynomial division. Here’s a step-by-step approach using synthetic division:
Step 1: Set up the division
We want to divide ( a^3 – 4 ) by ( a + 2 ).
Step 2: Apply the Remainder Theorem
According to the Remainder Theorem, the remainder of a polynomial ( f(a) ) divided by ( a – c ) is given by ( f(c) ). In this case, we are dividing by ( a + 2 ), which means ( c = -2 ).
To find the remainder, we substitute ( a = -2 ) into the polynomial ( f(a) = a^3 – 4 ).
Step 3: Substitute ( a = -2 ) into the polynomial
Now, calculate ( f(-2) ):
[
f(-2) = (-2)^3 – 4 = -8 – 4 = -12
]
Thus, the remainder when ( a^3 – 4 ) is divided by ( a + 2 ) is -12.
Explanation
When dividing a polynomial by a linear factor like ( a + 2 ), the Remainder Theorem simplifies the process by allowing us to substitute ( a = -2 ) directly into the polynomial to find the remainder. Instead of performing long polynomial division or synthetic division, the Remainder Theorem gives us a quick shortcut.
In this case, we substituted ( a = -2 ) into ( f(a) = a^3 – 4 ) and found that the value of the polynomial at ( a = -2 ) is ( -12 ). This result is the remainder because the remainder theorem tells us that dividing a polynomial by ( a – c ) gives a remainder of ( f(c) ). Since we are dividing by ( a + 2 ), the remainder is ( f(-2) = -12 ).
Hence, the remainder is -12.