Class 9 Maths: Polynomials and Algebraic Identities Quiz

Maths Class 9 Ncert: Polynomials

Polynomials are expressions involving variables and coefficients. They are an essential topic in Class 9 Maths, as they form the basis for higher-order algebraic calculations. In this article, we will focus on the key subtopics of polynomials as described in the NCERT book.

Polynomials in One Variable

A polynomial in one variable is an expression of the form $$ax^n$$, where $$a$$ is a constant and $$n$$ is a non-negative integer. Polynomials can have any number of terms, and the highest power of the variable is called the degree of the polynomial. For example, the polynomial $$3x^2 + 2x + 1$$ has a degree of 2.

Zeroes of a Polynomial

The zeroes of a polynomial are the values of the variable for which the polynomial equals zero. Finding the zeroes of a polynomial is an essential skill in solving polynomial equations and understanding its graphical behavior.

Remainder Theorem

The remainder theorem states that when a polynomial $$f(x)$$ is divided by $$(x - a)$$, the remainder is either zero or a constant. This theorem is useful for factoring polynomials and finding their zeroes.

Factorization of Polynomials

Factorization is the process of breaking down a polynomial into simpler factors. The main aim of factoring is to find the roots (zeroes) of the polynomial and understand its graphical behavior. Factoring can be done using various methods, such as the factor theorem, grouping, and applying quadratic formulas.

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables involved. They are used to simplify expressions, solve equations, and verify Rolle's Theorem and Lagrange's Mean Value Theorem. Some common algebraic identities include $$(a + b)^2 = a^2 + 2ab + b^2$$, $$(a - b)^2 = a^2 - 2ab + b^2$$, and $$(a + b)(a - b) = a^2 - b^2$$.

Examples and Problems

To better understand the topics, let's solve some examples based on the subtopics discussed above:

1. Find the degree of the polynomial $$4x^3 - 3x^2 + 2x$$.

   The degree of a polynomial is the highest power of the variable in the expression. In this case, the degree is 3, as the term with the highest power of $$x$$ is $$4x^3$$.

2. Find the zeroes of the polynomial $$2x^2 - 4x + 3$$.

   To find the zeroes, we can use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$, where $$a = 2$$, $$b = -4$$, and $$c = 3$$. Plugging in the values, we get $$x = \frac{-4 \pm \sqrt{(-4)^2 - 4(2)(3)}}{2(2)}$$, which simplifies to $$x = \frac{-4 \pm \sqrt{16}}{4}$$, and finally, $$x = \frac{-4 \pm 4}{4} = -1$$, 2.

3. Factorize the polynomial $$6x^2 - 11x + 4$$.

   To factorize this polynomial, we can use the quadratic formula and find the roots: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$, where $$a = 6$$, $$b = -11$$, and $$c = 4$$. Plugging in the values, we get $$x = \frac{-11 \pm \sqrt{(-11)^2 - 4(6)(4)}}{2(6)}$$, which simplifies to $$x = \frac{-11 \pm \sqrt{121}}{12}$$, and finally, $$x = \frac{-11 \pm 11}{12} = \frac{0}{12}$$, $$\frac{11}{12}$$.

After factorizing, we can find the roots and use the root theorem to find the zeroes of the polynomial.

In conclusion, understanding the subtopics of polynomials in Class 9 Maths is crucial for mastering algebra and solving more complex problems. Practicing with examples and problems based on these subtopics will help strengthen your conceptual understanding and problem-solving skills.