1. Determine the remainder when 9x5 – 4x4 is divided by 3x-1
2. Show, using factor theorem, that 2x-1 is a factor of:
2x4 - x3-6x2 + 5x -1
and hence express 2x4 - x3-6x2 + 5x -1 as a product of a linear and cubic factor
Where do I start?
You can use either the polynomial long division, or the polynomial remainder theorem. According to the latter, the remainder of f(x) divided by k(x-a) is f(a).1. Determine the remainder when 9x5 – 4x4 is divided by 3x-1
According to the polynomial remainder theorem above (in this case it is the same as the factor theorem), it is sufficient to represent 2x-1 as $\displaystyle k(x - a)$ for some $\displaystyle k$ and $\displaystyle a$ and then to show that $\displaystyle a$ is the root of the given polynomial. To find the cubic factor, use the long division to divide the given fourth-degree polynomial by 2x - 1.2. Show, using factor theorem, that 2x-1 is a factor of:
2x4 - x3-6x2 + 5x -1
and hence express 2x4 - x3-6x2 + 5x -1 as a product of a linear and cubic factor
1. $\displaystyle \displaystyle f(x) = 9x^5 - 4x^4$.
The remainder when divided by $\displaystyle \displaystyle 3x - 1$ is given by $\displaystyle \displaystyle f\left(\frac{1}{3}\right)$.
2. $\displaystyle \displaystyle f(x) = 2x^4 - x^3 - 6x^2 + 5x - 1$.
To show that $\displaystyle \displaystyle 2x - 1$ is a factor, check that $\displaystyle \displaystyle f\left(\frac{1}{2}\right) = 0$.
Then you need to use long division to express $\displaystyle \displaystyle f(x)$ as $\displaystyle \displaystyle (2x-1)Q(x)$.