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Polynomials
1.0Master Degrees, Graphical Zeros, and Symmetric Functions in Minutes
Unlock the algebraic structure of polynomial expressions. Learn how to classify expressions by terms and degrees, discover the exact geometric meaning of polynomial roots, master the structural links connecting roots to coefficients, and apply division algorithms alongside symmetric functions to ace your Class 10 board exams.
An algebraic expression of the form , where (i) (ii) are real numbers (iii) power of x is a non negative integer, is called a polynomial.
Where , are coefficients of respectively and are terms of the polynomial. Here the term is called the leading term and its coefficient is the leading coefficient. For e.g.: is a polynomial in variable . and -4 are known as terms of polynomial and and 2 are their respective coefficients.
- All algebraic expressions are not polynomials.
4.0Types of polynomials:
Generally we divide the polynomials in three categories.
Polynomials classified by number of distinct variables
Generally, a polynomial in more than one variable is called a multivariate polynomial. A second major way of classifying polynomials is by their degree.
- Recall that the degree of a term is the sum of the exponents on variables, and that the degree of a polynomial is the largest degree of any one term. e.g. Polynomial here, degrees of term are respectively. Degree of given polynomial Polynomials classified by degree
Usually, a polynomial of degree , for greater than 3 , is called a polynomial of degree , although the phrases quartic polynomial and quantic polynomial are sometimes used for degree 4 and 5 respectively. Polynomials classified by number of non-zero terms
If a polynomial has only one variable, then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers").
5.0Value of a polynomial
If is a polynomial in variable and is any real number, then the value obtained by replacing by a in is called value of at and is denoted by . For example : Find the value of at Zero of a polynomial : A real number is a zero of the polynomial if . For example: Consider Thus, 1,2 and 3 are called the zeros of polynomial .
6.0Geometrical meaning of the zeros of a polynomial
Geometrically the zeros of a polynomials are the x -co-ordinates of the points where the graph intersects -axis. To understand it, we will see the geometrical representations of linear and quadratic polynomials. Geometrical representation of the zero of a linear polynomial
The graph of intersects the -axis at only one point . Therefore, we conclude that the linear polynomial has one and only one zero. Geometrical representation of the zero of a quadratic polynomial
Consider the following cases:
Case-I : Here, the graph cuts x -axis at two distinct points A and A'. (see figure)
The -coordinates of and are two zeros of the quadratic polynomial
Case-II : Here, the graph touches the x -axis at exactly one point (see fig). So, the two points A and A' of Case-I coincide here to become one point A. The x -coordinate of A is the only zero for the quadratic polynomial in this case.
Case-III : Here, the graph is either completely above the x -axis or completely below the x -axis (see fig.). So, it does not cut the x -axis at any point. So, the quadratic polynomial has no zero in this case.
So, you can see geometrically that a quadratic polynomial can have either two distinct zeros or one zero, or no zero. This also means that a polynomial of degree 2 has at most two zeros.
- In general, given a polynomial of degree , the graph of intersects the x -axis at atmost n points. Therefore, a polynomial of degree n has at most n zeros.
Building Concepts 1 The graphical representation of is shown in the given figure. Does it have (i) any solution (ii) zero(s)
Building Concepts 2 Look at the graphs given below. Each is the graph of , where is a polynomial. For each of the graphs, find the number of zeros of .
7.0Relationship between the zeros and coefficients of a polynomial
For a linear polynomial , we have, zero of a linear polynomial For a quadratic polynomial , with and as it's zeros, and are the factors of .
Therefore, , (where is a constant to balance the equation of the coefficient of i.e. .) comparing the coefficients of and constant terms on both the sides, we get and
This gives Sum of zeros Product of zeros If and are the zeros of a quadratic polynomial . Then polynomial is given by or sum of the zeros product of the zeros
The cubic polynomial whose zeros are and is given by
- If a polynomial has no real zero, it does not mean that it has no solution.
Numerical Ability 1 Find the zeros of the quadratic polynomial and verify the relation between the zeros and its coefficients. Solution: We have, For a cubic polynomial with and as its zeros, we have Thus, the zeros of are and Now,sum of the zeros and Sum of the zeros Product of the zeros and, Product of the zeros
Numerical Ability 2 Find the zeros of the quadratic polynomial and verify the relationship between the zeros and its coefficients. Solution: So, the value of is zero when or , i.e. or Therefore, and are the zeros of . Now, sum of zeros Product of zeros
Numerical Ability 3 Find a quadratic polynomial whose zeros are 5 and -2 respectively. Solution: Let the quadratic polynomial be , and it's zeros be and we have, We know that a quadratic polynomial when the sum and product of its zeros is given by sum of zeros product of zeros] where is a constant so,
Numerical Ability 4 Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively: (i) (ii) We know that a quadratic polynomial when the sum and product of its zeros are given by (Sum of the zeros) Product of the zeros , where is a constant. (i) Required quadratic polynomial is given by (ii) Required quadratic polynomial is given by
8.0Division Algorithm for polynomials
If is a polynomial and is a non-zero polynomial, then there exist two polynomials and such that , where or degree degree . In other words,
Dividend = Divisor \times Quotient + Remainder
Remark : If , then polynomial is a factor of polynomial .
Building Concepts 3 Check whether the polynomial is a factor of the polynomial , by dividing the second polynomial by the first polynomial. Explanation: We have
Building Concepts 4 Find all the zeros of , if you know that two of its zeros are and . Explanation: Let be the given polynomial. Since two zeros are and so, and are both factors of the given polynomial . Also, is a factor of the polynomial . Now, we divide the given polynomial by .
Numerical Ability 5 Divide the polynomial by the polynomial and verify the division algorithm. Solution: We have
9.0Symmetric functions of the zeros
Let be the zeros of a quadratic polynomial, then the expression of the form ; are called the symmetric functions of the zeros. By symmetric function we mean that the function remains invariant (unaltered) in values when the roots are changed cyclically. In other words, an expression involving and which remains unchanged by interchanging and is called a symmetric function of and .
Some useful relations involving and are :- (i) (ii) (iii) (iv) (v)
Numerical Ability 6 If and are the zeros of the quadratic polynomial then calculate (i) (ii) Solution Since and are the zeros of the quadratic polynomial (i) We have, (ii) We have,
Numerical Ability 7 If and are the zeros of the quadratic polynomial , find the value of Solution: Since and are the zeros of the polynomial . and We have
Numerical Ability 8 If and are the roots (zeros) of the polynomial such that , find the value of . Solution: Since and are the roots (zeros) of the polynomial . Hence, the value of k is 2 .
Numerical Ability 9 If are the zeros of the polynomial satisfying the relation , then find the value of for this to be possible. Solution Since and are the zeros of the polynomial . and Now, and
10.0Important topics in Class 10 Maths: Polynomials
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12.0Supporting Study Materials
This study material, containing comprehensive CBSE Notes and NCERT Solutions for Chapter 2 of Class 10 Maths, is structured according to the latest NCERT guidelines. Featuring clear parabolic graph models, step-by-step polynomial long division blocks, and symmetric equation lookup tables, this guide provides complete preparation for your school assessments and board exams.
30-Second Quick Review: Polynomials
- A polynomial is an algebraic expression with non-negative integral powers of variables.
- The highest power of the variable is called the degree of the polynomial.
- A linear polynomial has one zero.
- A quadratic polynomial has at most two zeros.
- A cubic polynomial has at most three zeros.
- For ax² + bx + c:
- Sum of zeros = –b/a
- Product of zeros = c/a
- The graph of a quadratic polynomial is a parabola.
- Division Algorithm: Dividend = Divisor × Quotient + Remainder
- The degree of the remainder is always less than the degree of the divisor.
13.0Previous Year Questions (PYQs) on Polynomials
Question Find the zeros of the polynomial: x² – 5x + 6
Answer Factorizing,
x² – 5x + 6
= (x – 2)(x – 3)
Therefore,
Zeros = 2 and 3
14.0Recommended Next Topics
- Coordinate Geometry
- Triangles
- Introduction to Trigonometry
- Applications of Trigonometry