Question

Write three more polynomials and create three questions for each of them.

Answer 1

Polynomials

Definition. An expression of the form

$\quad a_{0}x^{n}+a_{1}x^{n-1}+...+a_{n-1}x+a_{n}$,

where $a_{0},\:a_{1},\:...,\:a_{n}$ are given numbers (real or complex), $n$ is a non-negative integers and $x$ is a variable, is called a polynomial in $x$ or a rational integral function of $x$.

Writing three polynomials and questions related to them:

1. Find the remainder when $4x^{5}+3x^{3}+6x^{2}+5$ is divided by $2x+1$. Here $4x^{5}+3x^{3}+6x^{2}+5$ is a polynomial in $x$.

2. Expand $f(x)=x^{4}-4x^{3}+3x^{2}+3x+7$ as a polynomial in $x-1$. Here $f(x)=x^{4}-4x^{3}+3x^{2}+3x+7$ is a polynomial in $x$.

3. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

• (i) $4x^{2}-3x+7$
• (ii) $y^{2}+\sqrt{2}$
• (iii) $3\sqrt{t}+t\sqrt{2}$
• (iv) $y+\frac{2}{y}$
• (v) $x^{10}+y^{3}+t^{50}$

Answer 2

Answer:

The three polynomials are:-

$3 {y}^{2} + 2y + 3 = 5$

$2y + 6 = 12$

$4 {y}^{2} + 2 {y}^{2} + 6y + 3 = 0$