Exercise 2.1

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

1. \(4x^2-3x+7\quad\Rightarrow\quad\) is a polynomial in one variable as there is only one variable \(x\)


2. \(y^2 + \sqrt{2}\quad\Rightarrow\quad\) is a polynomial in one variable as it contains only one variable \(y\)


3. \(3\sqrt{t} + t\sqrt{2}\quad\Rightarrow\quad\) is not a polynomial as variable \(t\) has 1/2 as an exponent. A polynomial must have exponents as whole numbers.



4. \(y + \frac{2}{y}\quad\Rightarrow\quad\) is a polynomial of one variable as it contains only \(y\) as variable



5. \(x^{10} + y^3 + t^{50}\quad\Rightarrow\quad\\\) is not a polynomial of one variable as it contains three variables \(x,~y ~\& ~t\)

3. Write the coefficients of \(x^2\) in each of the following:

1. \(2 + x^2 +x\quad\Rightarrow\quad\) Coefficient of \(x^2\) is 1


2. \(2 - x^2 +x^3\quad\Rightarrow\quad\) Coefficient of \(x^2\) is -1


3. \(\frac{\pi}{2}x^2 +x\quad\Rightarrow\quad\) Coefficient of \(x^2\) is \(\frac{\pi}{2}\)


4. \(\sqrt{2}x-1\quad\Rightarrow\quad\) Coefficient of \(x^2\) is 0 as polynomial can be wriiten as \(\Rightarrow 0\cdot x^2+\sqrt{2}x-1\)

3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.

1. \(x^{35} + 1\)


2. \(2x^{100}\)

4. Write the degree of each of the following polynomials:

1. \(5x^3 + 4x^2 + 7x\quad\Rightarrow\quad\) Degree = 3 (Highest power of x is 3)


2. \( 4 – y^2\quad\Rightarrow\quad\) Degree = 2 (Highest power of y is 2)


3. \( 5t – \sqrt{7}\quad\Rightarrow\quad\) Degree = 1 (Highest power of t is 1)


4. \( 3\quad\Rightarrow\quad\) Degree = 0 (3 ia a constant and the degree of a constant is always zero)

5. Classify the following as linear, quadratic and cubic polynomials:

1. \(x^2 + x\quad\Rightarrow\quad\) Quadratic Polynomial, Highest power of \(x\) is 2


2. \(x – x^3\quad\Rightarrow\quad\) Cubic Polynomial, Highest power of \(x\) is 3


3. \(y + y^2 + 4\quad\Rightarrow\quad\) Quadratic Polynomial, Highest power of \(y\) is 2


4. \(1+x\quad\Rightarrow\quad\) Linear Polynomial, Highest power of \(x\) is 1


5. \(3t\quad\Rightarrow\quad\) Linear Polynomial, Highest power of \(t\) is 1


6. \(r^2\quad\Rightarrow\quad\) Quadratic Polynomial, Highest power of \(r\) is 2


7. \(7x^3\quad\Rightarrow\quad\) Cubic Polynomial, Highest power of \(x\) is 3