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Q4(i):
Write the degree of each of the following polynomials: (i) $5x^3 + 4x^2 + 7x$

Solution :

Step 1: Initial Setup & Theoretical Foundation

Let the given mathematical expression be defined as a polynomial function $P(x)$ in terms of the single variable $x$:

$P(x) = 5x^3 + 4x^2 + 7x$

[Per the Fundamental Theorem of Algebra and standard algebraic definitions, the degree of a polynomial in one variable is defined as the highest exponent (power) of the variable present in the expression, provided that the coefficient of that term is strictly non-zero.]

Step 2: Term-by-Term Exponent Analysis

To determine the degree, we must decompose the polynomial into its constituent terms and isolate the exponent of the variable $x$ in each term.

  • First Term: $5x^3$
    The variable is $x$, and its exponent is $3$. The coefficient is $5$ (where $5 \neq 0$).
  • Second Term: $4x^2$
    The variable is $x$, and its exponent is $2$. The coefficient is $4$ (where $4 \neq 0$).
  • Third Term: $7x$
    By the laws of exponents, any variable without an explicitly written power has an implicit exponent of $1$. Thus, $7x$ can be rewritten as $7x^1$. The exponent is $1$. The coefficient is $7$ (where $7 \neq 0$).

Step 3: Visualizing the Polynomial Structure

The following diagram isolates each term, highlighting the exponents to visually confirm the maximum power.

5x3 + 4x2 + 7x1 Highest Exponent = Degree Exponent: 2 Exponent: 1

Step 4: Determination of the Maximum Exponent

We extract the set of all exponents present in the polynomial $P(x)$:

$E = \{3, 2, 1\}$

Comparing these values, we find the maximum element in the set:

$\max(3, 2, 1) = 3$

Because the coefficient associated with $x^3$ is $5$ (which satisfies the condition $5 \neq 0$), the highest power is validated as the degree of the polynomial.

Final Solution: The degree of the polynomial $5x^3 + 4x^2 + 7x$ is $3$.


More Questions from Class 9 Mathematics Polynomials EXERCISE 2.1


CBSE Solutions for Class 9 Mathematics Polynomials


Chapters in CBSE - Class 9 Mathematics


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