Embibe Experts Solutions for Chapter: Polynomials, Exercise 1: Exercise

A Maths teacher was teaching 'Polynomials' to her students. She involved three of her students to make the topic interesting. The teacher gave them some values of a polynomial $P\left(x\right)$, such that, $P\left(0\right)=1, P\left(1\right)=0, P\left(2\right)=2$.

She asked them, "Which of the following is definitely a factor of $P\left(x\right)$?"

Geetha selected randomly two polynomials from the box consists of many polynomials. The selected polynomials are $P\left(x\right)=2x^3-x^2-5x+11$ and $G\left(x\right)=x+1$. She divided the polynomial $P\left(x\right)$ by $G\left(x\right)$, then she found that $P\left(x\right)$ is not divisible by $G\left(x\right)$. Geetha's father asked her to add some number to $P\left(x\right)$  such that resultant polynomial is completely divisible by $G\left(x\right)$. Find out the number to be added to $P\left(x\right)$ to make it is divisible by $G\left(x\right)$.

A shipment service provider uses three types of containers for shipping materials. The height and width of the three containers are the same. The containers' width is $0.5 \mathrm{m}$ more than their height, and the volume of the smallest container is $500 {\mathrm{m}}^3$.

Which of the following statements is true?

In 2002 NatWest trophy final against England, Sourav Ganguly-led India script history with NatWest Trophy triumph. If product of runs scored by Sourav Ganguly, Mohammad Kaif and Yuvraj Singh is given by the expression $x^3-15x^2+50x$, then the sum of total runs scored by Sourav Ganguly, Mohammad Kaif and Yuvraj Singh in terms of $x$ will be

Two students in class of IX named Manasa and Uma were assigned a polynomial by their maths teacher. The polynomial was $p\left(x\right)=x^2-7x+12$.

They were asked to express this polynomial as product of factors. Both applied factorisation by splitting the middle term and got different answer.

Manasa's answer: $\left(x-3\right)\left(x-4\right)$

Uma's answer: $\left(x+3\right)\left(x-4\right)$.

Find out whose answer is correct and show factorisation.

Two brothers Gokul and Gowtham decided to build a house. They decided to share the amount depending on the variable cost of materials. The amount shared by two brothers is given by the expression $\left(6x^2+11x-35\right)$, which is the product of their individual share factors. Find the value of $x$ if their shares are equal.

Rishav donated a certain amount of money to an NGO. His friend Pankaj wanted to know the amount donated by him. But, Rishav did not disclose the amount he donated and instead gave him a hint that if $\left(x+\frac{1}{x}\right)=₹9$, then the amount donated by him is $₹\left(x^3+\frac{1}{x^3}\right)$. Find the amount donated by Rishav to the NGO in $₹$

In a section, a student had his birthday, and decided to take a box of chocolates for his classmates. Let the box of chocolates be represented by the polynomial $f\left(x\right)=x^4-2x^3+3x^2-ax+b$. On the way to school, he calculated that, if he distributes all the chocolates equally between $\left(x+1\right)$ classmates, then $19$ toffees are left with him. Whereas, if he leaves out $2$ boys who were bad to him, and distributes it equally between $\left(x-1\right)$ friends, then $5$ chocolates will be left. After he reached school, he found that $4$ of his classmates were absent that day, and he distributed all the chocolates equally amongst $\left(x-3\right)$ friends. Find out the number of chocolates that were left with him now.