Polynomial Division (1 Viewer)

[Speed6]

Just wanting a solution guide through this question, especially where there is a part later in the working out involving fractions. So can you please show me where these decimals or fractions came and how.

Thanks, will rep if available

 

[Drongoski]

One way:

6x^2 - 3x + 1 = 2x(3x-2) + (x+1)

 

[SilentWaters]

A continuation of Drogonski's response - dividing the right-hand side by 3x-2, we get

(*)​

Looking at the fraction, we can express it as



Now the second factor here can be manipulated into



so putting it all together in (*), we end up with



Note that we now have a fraction whose numerator cannot be further divided by the denominator.

Because it has a lower degree/power of x it forms part of the remainder. We expand it out to obtain the final form:



and so

 

[SilentWaters]

Alternatively, we find R(x) first, using the remainder theorem: "the remainder of the division of a polynomial f(x) by some linear polynomial bx-a is equal to f(a/b)".

P(2/3) = 5/3, so that's our remainder.

We take the remainder away from what we're dividing to find what's exactly divisible by 3x-2.

From here, we can determine the exact quotient, Q(x), which must be of the form Bx+A since the highest power of x in P(x) is 2:



By equating the coefficients of x squared and the constant terms, we can easily see that (1) 6 = 3B and (2) -2/3 = -2A.

Hence B = 2 and A = 1/3.

Putting it together:

 

[Drsoccerball]

Or the traditional method of long division using fractions in your divisions