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Q4(ii):
Write the degree of each of the following polynomials:
(ii) $4 – y^2$
Solution :
Initial Setup & Polynomial Definition
We are given the following algebraic expression in terms of the variable $y$:
$P(y) = 4 - y^2$
[Per the fundamental theorem of algebra and polynomial definitions], a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Step 1: Standard Form Representation
To rigorously analyze the polynomial, we first rewrite it in its standard form. The standard form of a single-variable polynomial dictates that the terms must be ordered in descending powers of the variable.
$P(y) = -y^2 + 4$
Next, we express every term explicitly with the variable $y$ to expose all hidden exponents. A constant term, such as $4$, can be written as $4y^0$ [since $y^0 = 1$ for all $y \neq 0$].
$P(y) = -1 \cdot y^2 + 4 \cdot y^0$
Step 2: Exponent Analysis & Term Breakdown
We now isolate and evaluate the exponent of the variable $y$ in each distinct term of the polynomial.
| Term | Coefficient | Variable Part | Exponent (Power) |
|---|---|---|---|
| $-y^2$ | $-1$ | $y^2$ | $2$ |
| $4$ | $4$ | $y^0$ | $0$ |
Step 3: Determination of the Degree
The degree of a polynomial in one variable is defined as the highest power (exponent) of the variable that appears with a non-zero coefficient.
- The exponent of the first term is $2$.
- The exponent of the second term is $0$.
Comparing the exponents: $\max(2, 0) = 2$.
Because the highest exponent is $2$, the polynomial is classified as a quadratic polynomial.
Geometric Visualization: The Quadratic Curve
To provide a comprehensive understanding, we can visualize the polynomial by plotting the relation $x = 4 - y^2$. Because the degree is $2$, the geometric representation is a parabola. Since the squared variable is $y$ and its coefficient is negative, the parabola opens to the left, with its vertex at the maximum $x$-value.
Final Solution: The degree of the polynomial $4 - y^2$ is 2.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.1
- Q1(i): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (i) $4x^2 – 3x + 7$
- Q1(ii): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (ii) $y^2 + \sqrt{2}$
- Q1(iii): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iii) $3\sqrt{t} + t\sqrt{2}$
- Q1(iv): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iv) $y + \frac{2}{y}$
- Q1(v): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (v) $x^{10} + y^3 + t^{50}$
- Q2(i): Write the coefficients of $x^2$ in each of the following: (i) $2 + x^2 + x$
- Q2(ii): Write the coefficients of $x^2$ in each of the following: (ii) $2 – x^2 + x^3$
- Q2(iii): Write the coefficients of $x^2$ in each of the following: (iii) $\frac{\pi}{2}x^2 + x$
- Q2(iv): Write the coefficients of $x^2$ in each of the following: (iv) $\sqrt{2}x - 1$
- Q3: Give one example each of a binomial of degree 35, and of a monomial of degree 100.
- Q4(i): Write the degree of each of the following polynomials: (i) $5x^3 + 4x^2 + 7x$
- Q4(iii): Write the degree of each of the following polynomials: (iii) $5t – \sqrt{7}$
- Q4(iv): Write the degree of each of the following polynomials: (iv) $3$
- Q5(i): Classify the following as linear, quadratic and cubic polynomials: (i) $x^2 + x$
- Q5(ii): Classify the following as linear, quadratic and cubic polynomials: (ii) $x – x^3$
- Q5(iii): Classify the following as linear, quadratic and cubic polynomials: (iii) $y + y^2 + 4$
- Q5(iv): Classify the following as linear, quadratic and cubic polynomials: (iv) $1 + x$
- Q5(v): Classify the following as linear, quadratic and cubic polynomials: (v) $3t$
- Q5(vi): Classify the following as linear, quadratic and cubic polynomials: (vi) $r^2$
- Q5(vii): Classify the following as linear, quadratic and cubic polynomials: (vii) $7x^3$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
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