Question

2) The remainder when dividing p(x) by x-1 yields a quotient of 1 and a remainder of -3.A) what is p(x)? what is the horizontal asymptote and y intercept of y = p(x) / x-1?

Answer 1

When we divide to polynomials, let's say p(x) and q(x), the result of this division can be written as:

`(p(x))/(q(x))=\text{quotient}+\frac{\text{reminder}}{q(x)}`

In this case we have q(x)=x-1, the quotient is 1 and the reminder -3 then we have:

`(p(x))/(x-1)=1-(3)/(x-1)`

So we have an equation for p(x). We can multiply both sides by x-1:

`\begin{gathered} (p(x))/(x-1)\cdot(x-1)=(1-(3)/(x-1))\cdot(x-1) \\ p(x)=(x-1)-(3)/(x-1)\cdot(x-1) \\ p(x)=x-1-3 \\ p(x)=x-4 \end{gathered}`

Then p(x)=x-4.

We still need to find the horizontal asymptote and y-intercept of p(x)/(x-1). This expression is:

`(x-4)/(x-1)`

Since both the numerator and the denominator have the same degree (i.e. the biggest power of x in both expressions is the same) then the horizontal asymptote is given by:

`y=(a)/(b)`

Where a and b are the leading coefficients of the numerator and the denominator respectively. The leading coefficient of a polynomial is the number multiplying the biggest power of x and in both cases this number is 1. Then a=1 and b=1 and the horizontal asymptote is y=1.

Finally the y intercept is the point (0,y) so we just need to take x=0 and solve:

`(0-4)/(0-1)=4`

Then the y-intercept is (0,4).