
Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time, and the product of its zeroes as 2, 7, 14 respectively.
Answer
519.6k+ views
Hint: A cubic polynomial is the polynomial whose degree is 3 and it has 3 roots. We will use the sum, sum of the products and products given in the question to find the cubic polynomial.
Complete step-by-step answer:
Now, according to the theory of the polynomials, if the cubic expression is
f(x) = and are the roots of this polynomial, then we have
sum of products = , where b is the coefficient of and a is the coefficient of .
Also, we have a sum of products taken two at a time = , where c is the coefficient of x.
Product of zeroes = , where d is a constant term.
Now, we are given that sum = 2. Therefore,
or b = -2a.
Also, the sum of product of zeroes taken two at a time = 7. Therefore,
or c = 7a.
Product of zeroes = 14. Therefore, or d = -14a.
Now, we have f(x) = . So, putting value of b, c and d in the expression, we get
f(x) =
Taking a common from the above expression, we get
f(x) =
So, the required polynomial is f(x) = .
Note: Whenever we come up with such types of questions, we will use the formula of sum, sum of product and product of zeroes to find any expression. Any polynomial expression can be found as we found the cubic polynomial. We have to follow a few steps to find the polynomial expression. First, we will let a polynomial expression, then we will use the formulas according to the values given in the question. Then we will find the values of variables using the formulas and put those values to get the desired polynomial expression.
Complete step-by-step answer:
Now, according to the theory of the polynomials, if the cubic expression is
f(x) =
sum of products =
Also, we have a sum of products taken two at a time =
Product of zeroes =
Now, we are given that sum = 2. Therefore,
Also, the sum of product of zeroes taken two at a time = 7. Therefore,
Product of zeroes = 14. Therefore,
Now, we have f(x) =
f(x) =
Taking a common from the above expression, we get
f(x) =
So, the required polynomial is f(x) =
Note: Whenever we come up with such types of questions, we will use the formula of sum, sum of product and product of zeroes to find any expression. Any polynomial expression can be found as we found the cubic polynomial. We have to follow a few steps to find the polynomial expression. First, we will let a polynomial expression, then we will use the formulas according to the values given in the question. Then we will find the values of variables using the formulas and put those values to get the desired polynomial expression.
Recently Updated Pages
Master Class 11 Physics: Engaging Questions & Answers for Success

Master Class 11 Chemistry: Engaging Questions & Answers for Success

Master Class 11 Biology: Engaging Questions & Answers for Success

Class 11 Question and Answer - Your Ultimate Solutions Guide

Master Class 11 Business Studies: Engaging Questions & Answers for Success

Master Class 11 Computer Science: Engaging Questions & Answers for Success

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Gautam Buddha was born in the year A581 BC B563 BC class 10 social science CBSE

Which one is a true fish A Jellyfish B Starfish C Dogfish class 10 biology CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

What is the median of the first 10 natural numbers class 10 maths CBSE
