Use the remainder theorem to find the reminder when f(x) is divide by x+3.

Question

Then use the factor theorem to determinewhether x+3 is a factor of f(x)

f(x)=3x^6-27x^4+x^2-5

Is x+3 a factor of f(x)=3x^6-27x^4+x^2-5

Answer 1

The Remainder Theorem:

When you divide a polynomial f(x) by x - a the remainder r will be f(a).

f(x) = 3x^6 - 27x^4 + x^2 - 5 is divided by (x + 3).

According to the Remainder Theorem: f( - 3) is the remaider.

f(- 3) = 3(- 3)^6 - 27(- 3)^4 + (- 3)^2 - 5

= (3*729) - (27*81) + 9 - 5

= 2187 - 2187 + 4

= 4.

4 is the Remainder.

Factor therom:

when f(c) = 0 then (x - c) is a factor of polynomial.

Let (x + 3) is a factor of f(x).

Then, f(- 3) = 3(- 3)^6 - 27(- 3)^4 + (- 3)^2 - 5

= (3*729) - (27*81) + 9 - 5

= 2187 - 2187 + 4

= 4.

Since 4 is not equals to zero(0), (x + 3) is a not a factor of f(x).