Ex.2.4 Q4 Polynomials Solution - NCERT Maths Class 10

Go back to  'Ex.2.4'


If two zeroes of the polynomial \({x^4}-6{x^3}-26{x^2} + 138x-35\) are \(2 \pm \sqrt 3 \) find other zeroes.

 Video Solution
Ex 2.4 | Question 4

Text Solution


What is known?

Two zeroes of the polynomial \(x^{4}-6 x^{3}-26 x^{2}+138 x-35\) are \(2 \pm \sqrt 3 .\)

What is unknown?

Other zeroes of the given polynomial.


Given polynomial and the zeroes of the polynomial are \(x^{4}-6 x^{3}-26 x^{2}+138 x-35\) and \(2 \pm \sqrt 3 .\)

By using the zeroes of a polynomial, you can find out the factor of the polynomial.

Now divide the polynomial with the factor, you will get the quotient and remainder of the polynomial.

Put this value in the division algorithm and you will get the other zeroes by simplifying its factors.


\[P\left( x \right) = {x^4}-6{x^3}-26{x^2} + 138x-35\]

Zeroes of the polynomial are \( = 2 \pm \sqrt 3 .\)

Therefore, \[\begin{align}\left( {x - 2 + \sqrt 3 } \right)\left( {x-2 - \sqrt 3 } \right) &= {{ }}{x^2} + 4-4x - 3\\&= {{ }}{x^2}-{{ }}4x{{ }} + {{ }}1\end{align}\]

is a factor of the given polynomial.

To find out the other polynomial, we have to find the quotient by dividing \({x^4}-6{x^3}-26{x^2} + 138x-35\) by \({x^2}-4x + 1\)

Clearly, by division algorithm,

\[{x^4}-6{x^3}-26{x^2} + 138x-35 = \left( {{x^2}-4x+ 1} \right)\left( {{x^2} - 2x- 35} \right)\]

It can be observed that \({x^2} - 2x - 35\) is a factor of the given polynomial and

\[\begin{align}{x^2} - 2x - 35 &= {x^2} - 7x + 2x - 35\\ &= \left( {x - 7} \right)\left( {x - 5} \right)\end{align}\]

Therefore, the value of the polynomial is also zero when \(x-7 = 0\) or \(x + 5 = 0\)

Or \(x=7\) and \(x=-5\)

Hence, 7 and -5 are also zeroes of this polynomial.

Learn math from the experts and clarify doubts instantly

  • Instant doubt clearing (live one on one)
  • Learn from India’s best math teachers
  • Completely personalized curriculum

Frequently Asked Questions

What are Class 10 NCERT Exemplars?
While getting good scores in school tests is a desirable outcome, it is not a reliable indicator of how strong your child’s math foundation really is. Many students who score well in school exams in their earlier years, might struggle with math in higher grades because of a weak foundation. At Cuemath, we evaluate your child’s grasp of math fundamentals, and take corrective actions immediately. Also, your child may have limited exposure in their school, and in most cases, may not feel challenged to learn more. Cuemath's customised learning plan ensures your child is challenged with varied difficulty levels of questions at every stage.
What is the difference between CBSE and NCERT syllabus for Class 10?
How will Class 10 NCERT books help in exam preparation?
How will Class 10 NCERT books help you understand basic math concepts?
Which is the best video solution for the class 10 maths NCERT?