Chapter 2: Polynomials and Factorisation (Exercise 2.1)

Chapter 2: Polynomials and Factorisation (Exercise 2.1)

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

1. $3x^2 - 2x + 5$

2. $x^2 + \sqrt{2}$

3. $p^2 - 3p + q$

4. $y + \frac{2}{y}$

5. $5\sqrt{x} + x\sqrt{5}$

6. $x^{10} + y^3 + t^{50}$

Solution:

1. $3x^2 - 2x + 5$:

   • Polynomial in one variable ($x$).

   • Reason: All exponents of the variable $x$ are non-negative integers ($2$ and $1$).

2. $x^2 + \sqrt{2}$:

   • Polynomial in one variable ($x$).

   • Reason: The exponent of $x$ is $2$, which is a non-negative integer ($\sqrt{2}$ is a constant).

3. $p^2 - 3p + q$:

   • Not a polynomial in one variable.

   • Reason: It contains two variables, $p$ and $q$.

4. $y + \frac{2}{y}$:

   • Not a polynomial.

   • Reason: Re-writing gives $y + 2y^{-1}$. The exponent of $y$ in the second term is $-1$, which is a negative integer.

5. $5\sqrt{x} + x\sqrt{5}$:

   • Not a polynomial.

   • Reason: Re-writing gives $5x^{\frac{1}{2}} + x\sqrt{5}$. The exponent of $x$ in the first term is $\frac{1}{2}$, which is a fraction (not a whole number).

6. $x^{10} + y^3 + t^{50}$:

   • Not a polynomial in one variable.

   • Reason: It involves three variables ($x, y,$ and $t$).

Question 2: Write the coefficient of $x^2$ in each of the following:

1. $2 + x^2 + x$

2. $2 - x^2 + x^3$

3. $\frac{\pi}{2}x^2 + x$

4. $\sqrt{2}x - 1$

5. $2x^2 - 3x + 5$

Solution:

1. $2 + x^2 + x$:

   • The term containing $x^2$ is $+1 \cdot x^2$.

   • Coefficient of $x^2$ = $1$

2. $2 - x^2 + x^3$:

   • The term containing $x^2$ is $-1 \cdot x^2$.

   • Coefficient of $x^2$ = $-1$

3. $\frac{\pi}{2}x^2 + x$:

   • The term containing $x^2$ is $\frac{\pi}{2}x^2$.

   • Coefficient of $x^2$ = $\frac{\pi}{2}$

4. $\sqrt{2}x - 1$:

   • Re-writing with $x^2$ term gives $0 \cdot x^2 + \sqrt{2}x - 1$.

   • Coefficient of $x^2$ = $0$

5. $2x^2 - 3x + 5$:

   • The term containing $x^2$ is $2x^2$.

   • Coefficient of $x^2$ = $2$

Question 3: Classify each of the following polynomials as linear, quadratic, cubic, or quartic:

1. $5x^2 + x - 7$

2. $x - x^3$

3. $x^2 + x + 4$

4. $x + 1$

5. $3t$

6. $r^2$

7. $2x^3 + 4x^2 + 5x + 7$

Solution:

1. $5x^2 + x - 7$: Degree = $2$ $\rightarrow$ Quadratic Polynomial

2. $x - x^3$: Degree = $3$ $\rightarrow$ Cubic Polynomial

3. $x^2 + x + 4$: Degree = $2$ $\rightarrow$ Quadratic Polynomial

4. $x + 1$: Degree = $1$ $\rightarrow$ Linear Polynomial

5. $3t$: Degree = $1$ $\rightarrow$ Linear Polynomial

6. $r^2$: Degree = $2$ $\rightarrow$ Quadratic Polynomial

7. $2x^3 + 4x^2 + 5x + 7$: Degree = $3$ $\rightarrow$ Cubic Polynomial

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

1. $5x^3 + 4x^2 + 7x$

2. $4 - y^2$

3. $5t - \sqrt{7}$

4. $3$

5. $0$

Solution:

1. $5x^3 + 4x^2 + 7x$: The highest power of $x$ is $3$. Degree = 3

2. $4 - y^2$: The highest power of $y$ is $2$. Degree = 2

3. $5t - \sqrt{7}$: The highest power of $t$ is $1$. Degree = 1

4. $3$: Can be written as $3x^0$. Degree = 0 (Non-zero constant polynomial)

5. $0$: The degree of the zero polynomial is Not Defined.

Question 5: Give one example each of:

1. A binomial of degree $35$.

2. A monomial of degree $100$.

Solution:

1. Binomial of degree $35$:

   • A polynomial having $2$ terms with highest degree $35$.

   • Example: $x^{35} + 7$

2. Monomial of degree $100$:

   • A polynomial having only $1$ term with degree $100$.

   • Example: $5y^{100}$