The remainder theorem states that when a polynomial p(x) is divided by a linear polynomial (x – a), then the remainder is equal to p(a) andThe factor theorem states that if f(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then, (x–a) is a factor of f(x), if f(a)=0.Find the remainder when p(x) = x3 − ax2 + 6x − a is divided by x − a.
Question
The remainder theorem states that when a polynomial p(x) is divided by a linear polynomial (x – a), then the remainder is equal to p(a) andThe factor theorem states that if f(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then, (x–a) is a factor of f(x), if f(a)=0.Find the remainder when p(x) = x3 − ax2 + 6x − a is divided by x − a.
Solution
To find the remainder when p(x) = x³ - ax² + 6x - a is divided by x - a, we can use the remainder theorem. According to the remainder theorem, the remainder is equal to p(a).
So, we substitute x = a in the polynomial p(x):
p(a) = a³ - aa² + 6a - a = a³ - a³ + 6a - a = 6a - a = 5a
So, the remainder when p(x) = x³ - ax² + 6x - a is divided by x - a is 5a.
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