Question

If 1 is a zero of the
polynomial,find the
remaining integral zeroes of the polynomial, if it has any.
​

Answer 1

Answer:

We’ve been talking about zeroes of polynomial and why we need them for a couple of sections now. We haven’t, however, really talked about how to actually find them for polynomials of degree greater than two. That is the topic of this section. Well, that’s kind of the topic of this section. In general, finding all the zeroes of any polynomial is a fairly difficult process. In this section we will give a process that will find all rational (i.e. integer or fractional) zeroes of a polynomial. We will be able to use the process for finding all the zeroes of a polynomial provided all but at most two of the zeroes are rational. If more than two of the zeroes are not rational then this process will not find all of the zeroes.

We will need the following theorem to get us started on this process.

So, why is this theorem so useful? Well, for starters it will allow us to write down a list of possible rational zeroes for a polynomial and more importantly, any rational zeroes of a polynomial WILL be in this list.

In other words, we can quickly determine all the rational zeroes of a polynomial simply by checking all the numbers in our list.

Before getting into the process of finding the zeroes of a polynomial let’s see how to come up with a list of possible rational zeroes for a polynomial

If the zeros of the polynomial are integers then the polynomial have integral zeros or integral solutions

We know that, if ‘a’ is the zero of the polynomial f(x) then f(a) = 0  

Given polynomial : f(y) = y3 – 2y2 + y + 4

Now,

Put y = – 1 in given a polynomial

⇒  f(– 1) = (– 1)3 – 2(– 1)2 + (– 1) + 4    = – 1 – 2 – 1 + 4 = 0

∴ (– 1) is the integral zero of f(y)

On dividing f(y) with (y + 1), we get the quotient as (y2 – 3y + 4)

⇒ f(y) = (y + 1)(y2 – 3y + 4)

Here, we cannot factorise (y2 – 3y + 4) but we get zeroes which are imaginary numbers.

∴ The given polynomial f(y) have only one integral solution

Step-by-step explanation:

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