# Number of positive integers x for which $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ , is a prime number is:

(A) $ 1 $

(B) $ 2 $

(C) $ 3 $

(D) $ 4 $

Answer

Verified

279.9k+ views

**Hint**: In the given question, we are given a function in the variable x and we have to find the number of positive integers x for which the function assumes a prime number value. So, we first factorize the given function into its factors. We use the factor theorem to find the factors of the cubic polynomial function. Then, we analyze the product of the prime factors in order to make the value of the function a prime number. A thorough understanding of functions, division algorithms and its applications will be of great significance.

**:**

__Complete step-by-step answer__In the given problem, we are required to find the number of positive integers for which the function $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ assumes a prime number value.

So, we first find the factors of the polynomial function $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ using the factor theorem.

According to the factor theorem, if the value of the variable obtained on equating the divisor to zero is a root of the polynomial, then the divisor polynomial is a factor of the dividend polynomial.

Now, let us substitute the value of the variable x as $ 1 $ in the function $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ .

So, we get,

$ \Rightarrow f\left( 1 \right) = {1^3} - 8{\left( 1 \right)^2} + 20\left( 1 \right) - 13 $

Simplifying the expression further, we get,

$ \Rightarrow f\left( 1 \right) = 1 - 8 + 20 - 13 $

$ \Rightarrow f\left( 1 \right) = 0 $

Since the value of polynomial function $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ for $ x = 1 $ is zero, then according to the factor theorem, $ \left( {x - 1} \right) $ is a factor of the polynomial function $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ .

Now, factoring the polynomial function to factor out $ \left( {x - 1} \right) $ , we get,

$ \Rightarrow f\left( x \right) = {x^3} - {x^2} - 7{x^2} + 7x + 13x - 13 $

$ \Rightarrow f\left( x \right) = {x^2}\left( {x - 1} \right) - 7x\left( {x - 1} \right) + 13\left( {x - 1} \right) $

Taking $ \left( {x - 1} \right) $ common from al the terms, we get,

$ \Rightarrow f\left( x \right) = \left( {{x^2} - 7x + 13} \right)\left( {x - 1} \right) $

Now, for the function $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ to be a prime number, one of the factors of the function must be $ 1 $ and the other factor must be a prime number.

So, there are two cases.

Case $ 1 $ : When $ \left( {x - 1} \right) = 1 $ and $ \left( {{x^2} - 7x + 13} \right) $ is a prime factor.

So, $ \left( {x - 1} \right) = 1 $

$ \Rightarrow x = 2 $

Hence, the value of x comes out to be $ 2 $ .

Now, we find the value of $ \left( {{x^2} - 7x + 13} \right) $ for $ x = 2 $ .

$ \Rightarrow {2^2} - 7\left( 2 \right) + 13 $

$ \Rightarrow 4 - 14 + 13 = 3 $

Therefore, $ x = 2 $ is a positive integral value of x for which the function $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ is a prime number.

Case $ 2 $ : When $ \left( {{x^2} - 7x + 13} \right) = 1 $ and $ \left( {x - 1} \right) $ is a prime factor.

So, $ \left( {{x^2} - 7x + 13} \right) = 1 $

$ \Rightarrow {x^2} - 7x + 12 = 0 $

Solving the quadratic equation using splitting the middle term method, we get,

$ \Rightarrow {x^2} - 3x - 4x + 12 = 0 $

$ \Rightarrow x\left( {x - 3} \right) - 4\left( {x - 3} \right) = 0 $

Taking $ \left( {x - 3} \right) $ common from the terms, we get,

$ \Rightarrow \left( {x - 4} \right)\left( {x - 3} \right) = 0 $

So, $ x = 3 $ or $ x = 4 $

Now, we find the value of $ \left( {x - 1} \right) $ for $ x = 3 $ and $ x = 4 $ .

So, $ \left( {x - 1} \right) = 2 $ for $ x = 3 $ .

Also, $ \left( {x - 1} \right) = 3 $ for $ x = 4 $ .

Therefore, $ x = 3 $ and $ x = 4 $ are the positive integral values of x for which the function $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ is a prime number.

Hence, the total number of positive integers for which $ f\left( x \right) = {x^3} - 8{x^2} + 20x - 13 $ is a prime number is $ 3 $ .

Therefore, option (C) is correct.

**So, the correct answer is “Option C”.**

**Note**: One must know the significance of the factor theorem in order to solve such problems. Substitution of a variable involves putting a certain value in place of the variable. That specified value may be a certain number or even any other variable. We must take care of the calculations while doing such problems. We must know the basics of the number system to solve such problems. Prime numbers are basic building blocks of arithmetic.

Recently Updated Pages

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

The ray passing through the of the lens is not deviated class 10 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

What is the nlx method How is it useful class 11 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the difference between anaerobic aerobic respiration class 10 biology CBSE