Class 9 Maths Chapter 2 Question Answers - Polynomials

Q1. In a school function, students having 100% attendance are to be honoured. Class teacher of IX A gives the number of eligible students aswhereas the class teacher of IX B gives the number of such students as If both the above numbers are equal then

(i) Find the number of prize-winners of 100% attendance in each section of class IX.
(ii) Which mathematical concept is used in the above problem?
(iii) By honouring 100% attendance holders, which value is depicted by the school administration?

(i) Number of eligible students of class IX A

=

Number of eligible students of class IX B

Since, the number of eligible students in both sections are equal.

Thus, in each section number of eligible students = 8
(ii) Number system.
(iii) Regularity.

Q2. Shivansh owns a rectangular garden that has area. He wants to estimate the length and width of the garden in order to estimate how many tree seeds he can sow along the boundary of the garden .
(i) Find the length and width of the garden.
(ii) Which mathematical concept is involved in this problem?
(iii) By trees what quality did Shivansh exhibit ?

(i) To find the length and width of rectangular garden simplify x²+9x−22 using middle term splitting method 
the product = ac=1×(−22)= −22
two numbers that multiply to ac=−22 and add up to $b\; =\; 9$b=9
Split the middle term (9x) into two terms:

x²+11x−2x−22
Group the terms and factorize:
=x(x+11)−2(x+11)
=(x−2)(x+11)
Therefore the length and width corresponds to : 
(x−2)(x+11)
(ii) Middle-term splitting of polynomials 
(iii) Shivansh exhibited the quality of environmental consciousness and responsibility. By planning to sow tree seeds along the boundaries, he shows a sense of care for the environment and an understanding of the importance of greenery. This reflects his commitment to sustainability and improving the surroundings while making optimal use of his garden space.

Q3. A group of (a + b) teachers, (a² + b²) girls and (a³ + b³) boys set out for an ‘Adult Education Mission’. If in the group, there are 10 teachers and 58 girls then:
(i) Find the number of boys.
(ii) Which mathematical concept is used in the above problem?

(iii) By working for ‘Adult Education’, which value is depicted by the teachers and students?

(i) ∵ (a + b)2 = a² + b² + 2ab
∴ 102 = 58 + 2ab [∵ Number of teachers = (a + b) = 10 and Number of girls = (a² + b²) = 58]
⇒ 100 = 58 + 2ab
⇒ 2ab = 100 – 58 = 42
⇒ ab = 42/2
= 21
Now, (a + b)³ = a³ + b³ + 3ab (a + b) ∴
(10)³ = a³ + b³ + 3 * 21 * 10
⇒ 1000 = a³ + b³ + 630 a³ + b³ = 1000 – 630 = 370
∴ Number of boys = 370.
(ii) Polynomials.
(iii) Social upliftment.

Q4. A school is organizing a fundraiser and plans to create rectangular stalls for selling items. The area of each stall is given by the polynomial x²+10x +21. The school principal wants to know the possible dimensions of the stalls.

(i) Factorize the polynomial to find the dimensions of each stall.

(ii) Which type of polynomial is mentioned in the question ?

(iii) What quality do the students exhibit through this initiative?

Sol: Factorizing the polynomial x²+10x +21

x²−10x+21
x²−7x−3x+21
Group the terms:
x(x−7)−3(x−7)
Factorize by taking the common binomial factor:
(x−7)(x−3)
Dimensions of each stall are: (x−7)(x−3)

(ii) The polynomial x²+10x+21 is a quadratic polynomial because its highest degree is 2.

(iii) The students exhibit teamwork, responsibility, and a sense of community service by organizing the fundraiser to contribute to a meaningful cause. They show dedication and collaboration to make the event successful.

Q5.  A community installs a rainwater harvesting tank in the shape of a rectangular shape. The volume of the tank is represented by the polynomial 3x²+11x+10, where $\mathrm{<br\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">}x$x represents the depth in meters.

(i) Factorize the polynomial to determine possible dimensions of the tank.

(ii) How can the community ensure efficient water storage?

(iii) What quality does the community display by installing this tank?

Sol: (i) Given: 3x²+11x+10 = 0
3x²+6x+5x+10 = 0
3x(x+2)+5(x+2)=0
(x+2)(3x+5)=0
x=−2,−5/3
(ii) Ensuring that storage tanks, reservoirs, and pipes are properly constructed and maintained prevents water wastage through leaks. Communities can focus on repairing old infrastructure and investing in modern, efficient water storage systems. By adopting this strategy, communities can improve water storage efficiency and make better use of available water resources.

(iii) By installing a water storage tank, the community displays responsibility, sustainability, and awareness, working together to conserve water and ensure its availability for the future.