Describe the vertical asymptote and holes for the graph of
y=(x-3)(x-1)/(x-1)(x-5)

Hole=1 and vertical asymptote=5    

Step-by-step explanation:

It is given that:

Look at the denominator and determine the values for which it is zero, therefore

(x-1)(x-5)=0⇒x=1,5

There is a common factor of (x-1), so we can cancel that common factor:

But note we can only do that provided x-1≠0, x≠1.

We have already got that when  x=1 the denominator is zero, so there is a removable discontinuity at that point (or a "hole").

when x=5, he denominator is also zero and this is the only vertical asymptote.

The prime polynomial or an irreducible polynomial is i.e., .

Further explanation:

In the question it is asked to choose the option which correctly signifies a prime polynomial.

The options are as follows:

Option A:

Option B:

Option C:

Option D:

A number which cannot be further factorized into the product of its factors is called a prime number. Just like a prime number there exists a prime polynomial.

A prime polynomial is also called as an irreducible polynomial.

An irreducible polynomial is a polynomial which cannot be expressed as product of its factors or a polynomial which cannot be factored into a polynomial of lower degree.

Option A:

In option A the polynomial is given as .

Consider the function as follows:

Substitute for in the above equation.

Therefore, the value of is .

As per the factor theorem is a factor of the polynomial .

Therefore, the polynomial has a factor as so, the polynomial given in the option A is not a prime polynomial.

This implies that option A is incorrect.

Option B:

In option B  the polynomial is given as .

Consider the function as follows:

Substitute for in the above equation.

Therefore, the value of is .

As per the factor theorem is a factor of the polynomial .

Therefore, the polynomial has a factor as so, the polynomial given in the option B is not a prime polynomial.

This implies that option B is incorrect.

Option C:

In option C the polynomial is given as .

Consider the function as follows:

Simplify the above expression as follows:

For the above function the degree or the highest power of the variable is . This implies that the above function is a quadratic function.

A quadratic function is a function in which the degree or the highest power of the variable is .

The general form of a quadratic function is as follows:

The discriminant of the above function is calculated as follows:

The quadratic function obtained above is as follows:

The discriminant of the above function is calculated as follows:

Since the discriminant is negative this means that there are no real roots of the function.

Since, the roots of the function are imaginary so, the function does not have any factor which would give a real root.

From the above statement it is concluded that the polynomial is a prime polynomial.

This implies that option C is correct.

Option D:

In option D the polynomial is given as .

Consider the function as follows:

Simplify the above expression as follows:

From the above calculation it is concluded that the function is factored as .

Therefore, the polynomial in not a prime polynomial.

This implies that option D is incorrect.

Thus, the prime polynomial or an irreducible polynomial is i.e., .

Learn more:

1.A problem on composite function  

2.A problem to find radius and center of circle  

3.A problem to determine intercepts of a line  

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Polynomial

Keywords: Polynomial, prime, prime polynomial, irreducible, irreducible polynomial, functions, factor, factor theorem, factorize, 10x2-5x+4x+6, quadratic, quadratic function, discriminant.