Your AI powered learning assistant

Polynomials Class 9

Intro

00:00:00

Polynomials are explained in a clear and accessible way, focusing on the foundational basics. The lesson integrates interactive videos, live classes, quizzes, and prompt feedback to enhance understanding. New CBSE courses in chemistry for class 9 and physics for classes 9 and 10 are now available with attractive discounts. Viewers are encouraged to subscribe, like, and share while joining a growing academic community.

Constants vs Variables

00:01:48

Constants are defined as fixed numerical values such as 6, 3, 2, -2, or even the square root of 2, remaining constant in any context. Variables, represented by letters like x, y, z or Greek letters such as alpha and beta, can change and assume different values. The explanation contrasts the unchanging nature of constants with the adaptability of variables, underscoring their distinct roles in polynomial expressions.

Algebraic Expression

00:03:04

Algebraic expressions consist of constants and variables interconnected by operations such as addition, subtraction, multiplication, and division. The critical observation is that only plus and minus signs separate distinct terms, while other operations merely combine elements within a term. In one example, 4x² + 5xy - 2 clearly has three terms because of these separators, whereas an expression using division without any plus or minus contains just one term. Understanding these distinctions allows for a precise evaluation of algebraic expressions.

Polynomials

00:06:15

Polynomials Defined by Whole Number Exponents An algebraic expression becomes a polynomial when every variable is raised to a non-negative integer exponent. Expressions like x³ - 4x² + 6x - 1 and those involving y with powers of 4 and 3 illustrate this condition clearly. Mastering this definition is essential for resolving multiple-choice questions and solidifying foundational algebra concepts.

Validating Polynomial Criteria Through Expression Analysis Examining candidate expressions reveals that an expression with a term like 1/x, which implies x raised to -1, violates the polynomial definition. A different example containing non-negative integer exponents for each term confirms its polynomial status. In contrast, an expression with a term equivalent to x^(1/2) does not satisfy the criteria, underscoring the need to verify each exponent carefully.

Coefficients

00:11:53

A polynomial's coefficient is the remaining factor after removing the specific variable part, and careful attention to the sign is essential. In the given example, the coefficient attached to x³ is 7, while the term involving y reveals a coefficient of -5x² once the variable is isolated. The same process shows that for a mixed term, extracting x² leaves -5y, the coefficient for x is 3, and the constant term is -2. This approach emphasizes that every remaining element, especially negative signs, must be included to avoid common errors.

Degree of a Polynomial

00:14:57

Polynomials in one variable designate their degree by the highest exponent, as seen when 2x + 5 highlights x raised to the first power. The concept adapts to polynomials with multiple variables by summing the exponents within each term, such as x²y² contributing a value of four. By comparing these sums, the term with the maximum total establishes the overall degree of the polynomial. Clear definitions and methodical evaluation make determining the degree intuitive even for complex expressions.

Polynomial Names based on Degree

00:18:44

Polynomials are classified by the highest power of x present in the expression. A first-degree polynomial, such as 2x + 5, is known as linear, while one of degree two, like x² + 3x - 4, is termed quadratic. A polynomial with degree three is recognized as cubic, and one with degree four is called bi quadratic, emphasizing how these names help distinguish their levels of complexity.

No of Terms in a Polynomial

00:20:21

Naming Polynomials by Term Count Polynomials are categorized by the number of distinct terms separated by addition or subtraction. Single-term expressions are called monomials, two-term expressions are binomials, and three-term expressions are trinomials. Constant expressions are considered constant polynomials, while the zero polynomial stands out as simply representing zero.

Determining the Degree: Constant versus Zero Polynomials A constant polynomial such as the number 2 is assigned a degree of 0 since it can be represented as 2 multiplied by x raised to the 0 power. The degree reflects the power of the variable present, so even when a variable does not visibly appear, the constant is understood to have degree 0. In contrast, the zero polynomial is unique because its degree is undefined, reflecting the ambiguity when it can be written with any power of x.

Zero of a Polynomial

00:25:04

A polynomial’s zero is the value of x that makes the polynomial equal to zero. For example, by substituting different values into the polynomial 2x-4, it is found that when x = 2 the expression becomes 0, confirming that 2 is the zero. Equating the polynomial to zero creates an equation, and by testing values through substitution, the correct solution is obtained.

Zero or Root of a Polynomial

00:28:56

A polynomial root is the value at which the polynomial equals zero, a concept distinct from the zero polynomial. Substituting specific values into the polynomial, like using x = 1 and x = -2 in x² + 3x + 5, results in outputs of 9 and 3 respectively. This clear demonstration of substitution highlights an efficient method for solving polynomial equations.

Q Find the zeroes of the polynomial

00:31:25

Deriving Roots Through Factorization The process begins by setting the quadratic polynomial equal to zero, transforming x² - x - 6 into the equation x² - x - 6 = 0. Factoring the expression reveals the factors (x + 2) and (x - 3). These factors directly yield the roots -2 and 3, highlighting the principle that zeros are the values making the polynomial vanish.

Extracting Zeros Using Common Factors The method continues with solving a quadratic expression by pulling out a common factor, as seen in x² - 3x = 0. Factoring results in the product x(x - 3) where each term can independently nullify the equation. This leads to the immediate identification of the zeros 0 and 3, confirming the efficiency of the factorization approach.

Verifying Polynomial Zeros Through Direct Substitution A verification technique is applied by substituting each candidate root back into the polynomial. For the equation x² - 5x + 6, inserting x = 2 and x = 3 yields zero in both cases. This substitution check serves as a robust method to confirm the accuracy of the factored roots and ensures the solution is correct.

Applying Concepts with Advanced Challenges and Resources An advanced challenge is introduced with the polynomial x² + 5x - 14, inviting further practice using the established methods. The problem encourages setting the equation to zero, factorization, and verification of results. Additional online resources and interactive sessions are highlighted as means to deepen understanding and enhance problem-solving skills.