These Class 9 Maths Notes and Chapter 2 Introduction to Linear Polynomials Class 9 Ganita Manjari Notes are designed according to the latest CBSE syllabus.
Class 9 Maths Chapter 2 Introduction to Linear Polynomials Notes
Class 9 Maths Ganita Manjari Chapter 2 Notes
Class 9 Introduction to Linear Polynomials Notes
→ Introduction: The word Polynomial is composed of two words Poly + Nomial, Poly = Many and Nomial = Terms. Which means an algebraic expression having many terms. To maintain unambiguity, we call algebraic expression having any number of terms a polynomial, provided the
variables must have degrees as whole numbers. For example, 2x, x3 + 12x – 1, x4 + 2x2 – 7, etc are polynomials, but 2x\(\frac{1}{3}\) – 1, x\(\frac{3}{2}\) + x + 12, x + \(\frac{1}{x}\), x2 + \(\frac{1}{x^3}\) – 1, etc are not polynomials.
→ An Algebraic Expression: An algebraic expression is a combination of numbers, variables and operation symbols, sometimes all at once or at least one at a time. For example, 2t2 + 3tk – 5k2 is an algebraic expression in the variables t and k.
→ Terms of an Algebraic Expression: In the algebraic expression 2t2 + 3tk – 5k2, 2t2, 3tk and – 5k2 are called the terms of the given algebraic expression.
→ Coefficients of the Terms: In the algebraic expression 2t2 + 3tk – 5k2, 2t2, 3tk and – 5k2 are called the terms of the given algebraic expression and 2, 3 and – 5 are the coefficients of the terms 2t2, 3tk, – 5k2, respectively.
→ Coefficient of the given Object: In the algebraic expression 2t2 + 3tk – 5k2, 2 is the coefficient of t2, 3t is the coefficient of k, 3 is the coefficient of tk and – 5 is the coefficient of k2.
→ Univa ate Polynomials: A polynomial having only one variable is called Univariate Polynomial. For example, 3x, 2x2 – 3x + 1, 3x3 – 3x2 + 7x + 10, etc are Univariate Polynomials.
→ Bivariate Polynomials: A polynomial having two variables is called Bivariate Polynomial. For example, x + y, x2 – xy, x3 – 2xy + y3 + 2y2 – x2y – 12, etc are Bivariate Polynomials as they are having two variables each.
→ Multivariate Polynomials: A polynomial having more than two variables is called Multivariate Polynomial. For example, x + y + z – 12, x2 – xy2 – x3z + 1, xy + yz + zx + 2, x + 3u – 3v – 2y2 + 2z + 4, etc are having more than two variables, so they are Multivariate Polynomials.
→ Degree of a Polynomial: The highest power of the variable in a polynomial is called its degree. For example, x2 – 12x + 7 has degree 2, y3 + 3y2 – 11y + 10 has degree 3, x2 – 2xy + 3y2 has degree 2, xyz + x2 – 2 has degree 3, etc.
→ Constant Polynomial: A polynomial having only constant term and no term containing letter – numbers called variables. For example, 2,3, – 7, written in context of a polynomial are constant polynomial.
Note: Constant polynomials have degree zero as the can be represented as 2x°, 3x°, – 7z°, etc.
→ Linear Polynomial: A polynomial of degree one is called a Linear Polynomial. Hence, 3x – 2,3 + 4y, 2 – 10, t + 4, 7u – 12, etc are linear polynomials in the variables x, y, z, t, u, respectively, x + y – 2z – 3 is also a linear polynomial as its degree is 1, though it has three variables.
Note: Polynomials are called functions also.
→ Quadratic Polynomial: A polynomial of degree 2 is called a Quadratic Polynomial. For example,
x2 – 3x + 2, y2 – 16, 2z2, etc are Quadratic Polynomials.
→ Cubic Polynomial: A polynomial of degree 3 is called a Cubic Polynomial. For example, x3 – 3x2 + 2, y2 – 16y3 – 2, 7z3, etc are Cubic Polynomials.
→ Monomial: A polynomial having only one term is called a Monomial. For example, 2t, 6x, 2xyz, 3x4, etc are monomials.
→ Binomial: A polynomial having two terms is called a Binomial. For example, a + b, 2x + y, x2 – y2, etc are binomials.
→ Trinomial: A polynomial having three terms is called a Trinomial. For example, a + b + c, 2x + y + 1, x2 – y2 + z2, etc are trinomials.
→ Value of a Polynomial: The value of a polynomial is obtained by substituting the given value of variable at which the value of the polynomial is required. For example, the value of the polynomial x2 – 3x + 2 at x = 1 is given by (1)2 – 3x(1) + 2 = 1 – 3 + 2 = 0 and the value of the same polynomial at x = 5 is given by (5)2 – 3 × (5) + 2 = 25 – 15 + 2 = 12.
Note: The value(s) of variable (letter – number) for which the value of any polynomial is zero, is called zeroes of the polynomial.
→ Linear Growth: It refers to a pattern in which a quantity increases by a fixed amount over equal intervals. For example, the sequence of numbers 2, 5, 8, 11, 14, …, has a linear growth .
→ Linear Decay: It refers to a pattern in which a quantity decreases by a fixed amount over equal intervals. For example, 67, 60, 53, 46, 39, …. has a linear decay.
→ Linear Pattern: A sequence of numbers where the difference between consecutive terms is constant is called a Linear Pattern. The example of linear growth or that of linear decay are the examples of Linear Patterns as well. So, 2, 5, 8, 11, 14, …, and 67, 60, 53, 46, 39, …, are examples of linear patterns also as they are examples of linear growth and linear decay, respectively.
→ Linear Relationship: A linear relationship between two variables x and y is represented by a straight line y = mx + c. The coefficient of x in this form of the equation is called the slope of this line. So, the slope of this line is ‘m’. The constant ‘c’ gives the distance of the point from the origin on y-axis at which the line intersects the y-axis, so ‘c’ is the y-intercept of the line.
Note: This form of equation of a straight line is called the slope-intercept form of a line. If the constant term is zero in this form of equation of a line, then it will pass through the origin. In this case the equation of the line becomes y = mx.
→ Parallel Lines: Two or more lines are said to be parallel lines if their slopes are same. For example, y = 4x + 12, y = 4x – 7, y = 4x + 5, y = 4x – √3, are all parallel lines as they are having same slope 4. So, we can write generally, y = mx + k, where m is a fixed constant and k can be any real number.
→ Another form of a Linear Relationship: ax + by + c = 0 is the other form of a linear relationship between the variables x and y.