NCERT Solutions for Class 9 Maths Chapter 2 Exercise 2.1

Understanding Polynomials in One Variable

An algebraic expression is called a polynomial if the exponents of the variables involved are non-negative integers (whole numbers). A polynomial in one variable, typically written as $P(x)$, contains only a single variable, such as $x$ or $y$.

Each part of the polynomial separated by a plus or minus sign is called a term. For example, in the expression $3x^2 + 5x - 7$, the terms are $3x^2$, $5x$, and $-7$. Every term has a real number multiplied by the variable, which is called its coefficient. In our example, the coefficient of $x^2$ is $3$, and the coefficient of $x$ is $5$. The number $-7$ is a constant term.

The degree of a polynomial is the highest exponent of the variable in that polynomial. For instance, the degree of $4x^3 - x^2 + 5$ is $3$. Knowing how to find degrees is critical because it tells us the fundamental behavior of the algebraic curve.

Core Classifications of Polynomials

Monomial

A polynomial containing exactly one non-zero term. Examples: $5x$, $-7$, or $3y^2$.

Binomial

A polynomial containing exactly two terms. Examples: $x + 1$, $y^2 - 9$.

Trinomial

A polynomial containing exactly three terms. Examples: $x^2 + 2x + 1$, $y^3 - y^2 + 4$.

Linear Polynomial

A polynomial of degree 1. Its general form is $ax + b$, where $a \neq 0$.

Quadratic Polynomial

A polynomial of degree 2. Its general form is $ax^2 + bx + c$, where $a \neq 0$.

Cubic Polynomial

A polynomial of degree 3. Its general form is $ax^3 + bx^2 + cx + d$, where $a \neq 0$.

Step-by-Step Guide to Solve Exercise 2.1

1. Check if the expression is a polynomial — Look at all the exponents of the variable. If any exponent is a fraction or a negative integer (e.g., $\sqrt{x} = x^{1/2}$ or $1/x = x^{-1}$), the expression is NOT a polynomial.
2. Determine the number of variables — Check if only one letter (like $x$, $y$, or $t$) is used as a variable. If you see multiple different letters (like $x^2 + y^3 + t^{10}$), it is a polynomial in more than one variable.
3. Identify coefficients — To find the coefficient of a term (e.g., $x^2$), look at the real number multiplied by it. Remember to include the negative sign if one is present before the term.
4. Find the degree of the polynomial — Locate the term with the highest power of the variable. The power of this term is the degree. For a non-zero constant term like $5$, the degree is $0$ because $5 = 5x^0$.

Common Exam Traps in Exercise 2.1

1. The Square Root Trap: Students often think $3\sqrt{t} + t\sqrt{2}$ is a polynomial because they see a square root. However, the variable $t$ is inside the square root ($t^{1/2}$), which has a fractional exponent. Hence, it is not a polynomial. (Note: $\sqrt{2}$ is a constant, which is fine, but the variable cannot be under a radical).

1. The Fraction Trap: An expression like $y + \frac{2}{y}$ is written as $y + 2y^{-1}$. Since the exponent of $y$ is $-1$ (not a whole number), this is not a polynomial.

1. Degree of Zero Polynomial: The degree of a non-zero constant polynomial (like $7$) is $0$. However, the degree of the zero polynomial (0) is not defined. Don't write $0$ as the degree of the zero polynomial!

Practice Questions with Solutions

• Q: State whether the expression $4y^2 - 5y + 11$ is a polynomial in one variable or not. Give reasons. A: Step 1: Analyze the terms of the expression. The terms are $4y^2$, $-5y$, and $11$. Step 2: Identify the variable. The only variable present is $y$, so it is in one variable. Step 3: Check the exponents of $y$. The powers of $y$ are $2$ (in $4y^2$), $1$ (in $-5y$), and $0$ (in $11$). Since $2, 1, 0$ are all non-negative integers (whole numbers), the exponents are valid. Final answer: Yes, $4y^2 - 5y + 11$ is a polynomial in one variable because the only variable is $y$ and all its exponents are whole numbers.
• Q: Find the coefficient of $x^2$ in the algebraic expression $2 - x^2 + x^3$. A: Step 1: Identify the term containing $x^2$. The term is $-x^2$. Step 2: Write down the numerical factor of $-x^2$. We can write $-x^2$ as $-1 \times x^2$. Step 3: Therefore, the coefficient is the multiplier, which is $-1$. Final answer: The coefficient of $x^2$ in $2 - x^2 + x^3$ is $-1$.
• Q: Find the degree of each of the following polynomials: (a) $5x^3 + 4x^2 + 7x$ (b) $5t - \sqrt{7}$ (c) $3$ A: Step 1: For (a) $5x^3 + 4x^2 + 7x$, the exponents of $x$ are $3$, $2$, and $1$. The highest exponent is $3$. Thus, the degree is $3$. Step 2: For (b) $5t - \sqrt{7}$, the exponent of $t$ is $1$. The highest exponent is $1$. Thus, the degree is $1$. Step 3: For (c) $3$, this is a non-zero constant polynomial. It can be written as $3x^0$. The highest exponent of the variable is $0$. Thus, the degree is $0$. Final answer: (a) Degree = 3, (b) Degree = 1, (c) Degree = 0.
• Q: Classify the following polynomials as linear, quadratic, or cubic: (i) $x - x^3$ (ii) $y + y^2 + 4$ (iii) $1 + x$ A: Step 1: Find the degree of polynomial (i) $x - x^3$. The highest exponent is $3$, so its degree is $3$. A polynomial of degree 3 is a cubic polynomial. Step 2: Find the degree of polynomial (ii) $y + y^2 + 4$. The highest exponent is $2$, so its degree is $2$. A polynomial of degree 2 is a quadratic polynomial. Step 3: Find the degree of polynomial (iii) $1 + x$. The highest exponent of $x$ is $1$, so its degree is $1$. A polynomial of degree 1 is a linear polynomial. Final answer: (i) Cubic, (ii) Quadratic, (iii) Linear.

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