When p(x) = x3 + ax2 + 2x + a is divided by (x + a), the remainder is
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To find the remainder when polynomial p(x) is divided by (x + a), we use the Remainder Theorem. This theorem states that the remainder of p(x) ÷ (x – c) is equal to p(c).
Here, our divisor is (x + a), which can be written as (x – (– a)). Therefore, c = – a. We substitute x = – a into the polynomial p(x) = x3 + ax2 + 2x + a.
Let's calculate p(– a):
p(− a) = (− a)3 + a(− a)2 + 2(− a) + a = − a3 + a(a2) − 2a + a = − a3 + a3 − 2a + a = − a
The terms – a3 and + a3 cancel each other out, leaving – 2a + a = – a. Therefore, the remainder is – a.
Final Answer: – a