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Q1(ii):
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (ii) $4s^2 - 4s + 1$

Solution :

Given: A quadratic polynomial $p(s) = 4s^2 - 4s + 1$.

To Find:
1. The zeroes of the polynomial $p(s)$.
2. Verification of the relationship between the zeroes and the coefficients of the polynomial.

Step 1: Finding the zeroes of the polynomial

To find the zeroes of $p(s)$, we set $p(s) = 0$.
$4s^2 - 4s + 1 = 0$

We use the splitting the middle term method. We look for two numbers whose product is $4 \times 1 = 4$ and whose sum is $-4$. These numbers are $-2$ and $-2$.

$4s^2 - 2s - 2s + 1 = 0$

Now, factor by grouping:
$2s(2s - 1) - 1(2s - 1) = 0$
$(2s - 1)(2s - 1) = 0$

Setting each factor to zero:
$2s - 1 = 0 \implies s = \frac{1}{2}$
$2s - 1 = 0 \implies s = \frac{1}{2}$

Thus, the zeroes of the polynomial are $\alpha = \frac{1}{2}$ and $\beta = \frac{1}{2}$.

Step 2: Identifying coefficients

Comparing $4s^2 - 4s + 1$ with the standard form $as^2 + bs + c$:
$a = 4$
$b = -4$
$c = 1$

Step 3: Verification of the relationship between zeroes and coefficients

The relationship states:
1. Sum of zeroes ($\alpha + \beta$) = $-\frac{b}{a}$
2. Product of zeroes ($\alpha \cdot \beta$) = $\frac{c}{a}$

Verification of Sum of Zeroes:
$\alpha + \beta = \frac{1}{2} + \frac{1}{2} = 1$
$-\frac{b}{a} = -\frac{(-4)}{4} = \frac{4}{4} = 1$
Since $1 = 1$, the sum of zeroes is verified.

Verification of Product of Zeroes:
$\alpha \cdot \beta = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
$\frac{c}{a} = \frac{1}{4}$
Since $\frac{1}{4} = \frac{1}{4}$, the product of zeroes is verified.

Final Answer: The zeroes of the polynomial $4s^2 - 4s + 1$ are $\frac{1}{2}$ and $\frac{1}{2}$. The relationship between the zeroes and coefficients is verified as the sum of zeroes is $1$ and the product of zeroes is $\frac{1}{4}$.

CBSE Solutions for Class 10 Mathematics Polynomials

Polynomials Class 10 EXERCISE 2.1

Chapters in CBSE - Class 10 Mathematics

Top Tutors who teach Polynomials

Bindu Davi

(51)

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Enamakkal, Enamavu

15 yrs of Exp

5 students

485 hrs of classes completed

500 per hour

Mathematics professional trainer with 15 years of teching experience in CBSE school.

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I started my teaching career in 2005 in that year itself I got a chance to take class in 10 for one month for leave vacancy. After that I got the chance in 2011 in another school, then it continued. Taking class in 10 it was a challenge because responsibilities are more. Both parents and students are in stress, at the same time the school management also. If the results are not satisfactory it will affect to the reputation of the school. So the teachers are alert always .All the students have to face the board. So the preparation of each and every child is important. For that conducting special classes ,morning classes. Conducting evaluations in proper interval. After every evaluation a session with parents in school is very important. Because they are more stressed than students. The year tenth is important because it is the turning point of each student . The effort of all the pupils will be fruitful when we get the good result. Teachers especially those who are taking in class 10 they are not bothered about the salary which the management as paying to them, of course it is a pain to their personal life. Once if we are standing in front of the students our main attention was that whether the concept will be reaching to the pupil in the right meaning and sense. In that time we are not bothered about the payment. If the student got the concept and scored good in exam that was the great reward for teacher. After the schooling that student and parents never forget that teacher in their life every important movements they expect that teachers presence there.

Usha

Very fortunate to find such an excellent teacher on Urbanpro. Bindu Davi very dedicated and clears the concepts.

Skills: Mathematics

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Virendra M.

(63)

Super Tutor

Sector 5, Dwarka, Delhi

10 yrs of Exp

13 students

1303 hrs of classes completed

600 per hour

10 Yrs in Teaching, Conduct Batch Of Two Kids, One-to-One classes. ShortTermPrep & LongTermClasses.

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I teach Math and Science to 8th-12th standard students and have been teaching for more than 10 years. Batch of 2 Students: I conduct these small batches with a pair of students of the same class and same education board (I cover all popular boards CBSE, ICSE, IGCSE etc.). This works for those students/parents who are looking for individual attention for their kids but still want it to be a little affordable. *3 classes per week (So 12 a month) *Maths and Science (Also English Grammar and Literature) *Fee Rs. 4500/- per month/12 classes. Batch of 6 Students: I conduct these small batches with 5-6 students of the same class and same education board (I cover all popular boards CBSE, ICSE, IGCSE etc.). *4 classes per week (So 16 a month) *Maths and Science *Fee Rs. 4000/- per month/16 classes. I will start the classes immediately once the first kid enrolls. -------------------------------------------------------------------------------Individual/Private/One-to-One classes: I conduct these one to one classes for those students willing to have a one-to-one setup (I cover all popular boards CBSE, ICSE, IGCSE etc.). *3 classes per week (So 12 a month) *Maths and Science (Also English Grammar and Literature) *Fee Rs. 7000/- per month/12 classes. I will start the classes immediately once the first kid enrolls. -------------------------------------------------------------------------------I took classes and batches in different institutions.

Puruzeet Singh

The experience with virendra sir was amazing. He did away with all my doubts and difficulties at all times , he is more than a teacher and a mentor to me. I'll always remember sir's debate's , scolding and care for me. Ty sir for making me a greater individual.

Skills: English , Mathematics and more

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