Class 10 Maths | 2018 Section C Q14

Zeroes of a polynomial. A standard text book question. However, it is a medium difficulty question because of the computation involved in arriving at the answer. Two zeroes of a polynomial are given and you have to find all the zeroes of the polynomial.

Question 14: Find all zeroes of the polynomial (2x^{4} – 9x^{3} + 5x^{2} + 3x – 1) if two of its zeroes are (2 + √3) and (2 - √3).

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Given polynomial is 2x^{4} – 9x^{3} + 5x^{2} + 3x – 1

Two of its zeroes: (2 + √3) and (2 - √3)

Because (2 + √3) and (2 - √3) are two zeros of the poynomial, (x – (2 + √3)) (x – (2 - √3)) is a factor of the given polynomial.

i.e., x^{2} - 2x + √3x - 2x + 4 - 2√3 - √3x + 2√3 - 3 is a factor of the polynomial.

Simplifying the above expression, we get x^{2} - 4x + 1 is a factor of the polynomial.

Because x^{2} - 4x + 1 is a factor of the polynomial, x^{2} – 4x + 1 will divide the polynomial without leaving a reaminder.

Let us do a long division of the polynomial by x^{2} – 4x + 1 and compute the quotient.

__ 2x ^{2} - x - 1 __

x

- x

-x

2x^{4} - 9x^{3} + 5x^{2} + 3x - 1 = (x^{2} - 4x + 1) (2x^{2} - x - 1)

2x^{2} - x - 1 can be factorized as 2x^{2} - 2x + x - 1

2x (x - 1) + 1(x - 1) = (2x + 1) (x - 1)

So, its zero are x = \-\frac{1}{2}) and 1

∴ The zeroes of the polynomial are (2 + √3), (2 - √3), -\\frac{1}{2}) and 1

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CBSE Class 10 Maths - 2022

- Real Numbers Revision Class
- Polynomials Revision Videos
- Linear Equations Revision Class
- Quadratic Equations Revision Class
- Arithmetic Progressions Revision Video
- Triangles Revision
- Coordinate Geometry Revision
- Trigonometry Revision Class
- Appl of Trigonometry
- Circles Revision Class
- Areas Related to Circles Revision Videos

1. Section A | 2018 CBSE Class 10 Maths 1 Mark **Questions 1 to 6**

2. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 7** | Real Numbers - Irrational Numbers

3. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 8** | Linear Equations in 2 variables

4. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 9** | Sum of Arithmetic Progressions

5. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 10** | Coordinate Geometry Section Formula

6. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 11** | Probability of Rolling 2 dice

7. Section B | 2018 CBSE Class 10 Maths 2 Mark **Question 12** | Probability of selecting numbers

8. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 13** | Real Numbers - LCM & HCF

9. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 14** | Polynomial Factorization

10. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 15A** | Coordinate Geometry - Parallelogram

11. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 15B** | Coordinate Geometry - Area of Triangles

12. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 16** | Quadratic Equations - Speed Time Distance

13. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 17** | Similar & Congruent Triangles

14. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 19** | Trigonometric Ratios

15. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 20** | Areas related to circles - sector area

16. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 21** | Surface Areas & Volumes - Cylinder, hemisphere and cone

17. Section C | 2018 CBSE Class 10 Maths 3 Mark **Question 22** | Statistics - Median

18. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 23** | Quadratic Equations _ Speed Distance Time

19. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 24** | Arithmetic Progressions

20. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 25A** | Equilateral & Right Triangles and Congruence

21. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 25B** | Pythagoram Theorem Proof

22. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 27** | Trigonometric Ratios & Identities

23. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 28** | Surface Area of Frustum

24. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 29** | Applications of Trigonometry - Heights & Distances

25. Section D | 2018 CBSE Class 10 Maths 4 Mark **Question 30** | Statistics - 2 Alternatives

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