Class 10 Exam  >  Class 10 Notes  >  Mathematics (Maths) Class 10  >  Important Definitions & Formulas: Polynomials

Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10 PDF Download

The chapter "Polynomials" is really important for learning the basics of algebra and how to use them to solve different math problems. It's like the foundation for solving all kinds of math questions in the future.

In this document, you'll find Class 10 Math Formulas related to Polynomials that can help you do well in your board exams and other important competitive exams.

Important Definitions

1. Polynomials

An algebraic expression of the form p(x) = a₀ + a₁ x + a₂ x2 + a₃ x3 + ...... + an xn, in which the variables involved have only non-negative integral exponents, is called a polynomial in x of degree n.

PolynomialPolynomial

Note:

In the polynomial a₀ + a₁ x + a₂ x2 + a₃ x3 + ...... + an xn

a₀, a₁ x, a₂ x2, a₃ x3  ..... an - 1 xn - 1, anxn are terms.

a₀, a₁, a₂, ..... an - 1, an are the co-efficients of x0, x1, x2, ....., xn-1, xn respectively.

2. Degree of a Polynomial

The highest power of the variable in a polynomial is called its degree.

Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10Example: 
5x + 3 is a polynomial in x of degree 1.
p(y) = 3y2 + 4y - 4 is a polynomial in y of degree 2.

  • Linear Polynomial: A polynomial of degree 1 is called a linear polynomial. A linear polynomial is generally written as ax + b (a ≠ 0), where a, and b are real coefficients.
  • Quadratic Polynomial: A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial is generally written as ax2 + bx + c (a ≠ 0), where a, b and c are real coefficients.
  • Cubic Polynomial: A polynomial of degree 3 is called a cubic polynomial. A cubic polynomial is generally written as ax3 + bx2 + cx + d (a ≠ 0), where a, b, c and d are real coefficients.

Examples:Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10

Question for Important Definitions & Formulas: Polynomials
Try yourself:The degree of the polynomial, x4 – x2 +2 is
View Solution

Value of a Polynomial at a Given Point 

If p (x) is a polynomial in x and ‘a’ is a real number. Then the value obtained by putting x = a in p (x) is called the value of p (x) at x = a.

Example: Let p(x) = 5x2 - 4x + 2 then its value at x = 2 is given by
p(2) = 5 (2)2 - 4 (2) + 2 = 5 (4) - 8 + 2 = 20 - 8 + 2 = 14
Thus, the value of p(x) at x = 2 is 14.

3. Zeroes of a Polynomial

A real number ‘a’ is said to be a zero of the polynomial p (x), if p (a) = 0.

Example: Let p (x) = x2 - x - 2 Then p (2) = (2)2 - (2) - 2 = 4 - 4 = 0,
and p (-1) = (-1)2 - (-1) - 2 = 2 - 2 = 0
∴ (-1) and (2) are the zeroes of the polynomial x- x - 2.

Note:

I. A linear polynomial has at the most one zero.
II. A quadratic polynomial has at the most two zeroes.

III. In general a polynomial of degree n has at the most n zeroes.

4. Geometrical Meaning of the Zeroes of a Polynomial


First, we consider a linear polynomial p (x) = ax + b.
Let ‘k’ be a zero, then p(k) = ak + b = 0
⇒ ak + b = 0
⇒ ak = - b orImportant Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10

The graph of a linear polynomial is always a straight line. It may or may not pass through the x-axis. In case the graph line is passing through a point on the x-axis, then the y-coordinate of that point must be zero. In general, for a linear polynomial ax + b = 0, (a ≠ 0), the graph is a straight line that can intersect the x-axis at exactly one point, namely, Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10 is the zero of the polynomial ax + b.
In the given figure, CD is meeting x-axis at x = -1.
∴ Zero of ax + b is -1.Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10

Note: 

A zero of a linear polynomial is the x-coordinate of the point, where the graph intersects the x-axis.

Question for Important Definitions & Formulas: Polynomials
Try yourself:What is the geometrical interpretation of the zeroes of a polynomial?
View Solution

Important Methods and Formulas

1. Graph of a Quadratic Polynomial

The graph of ax2 + bx + c, (a ≠0) is a curve of ∪ shape, called a parabola

  • If a > 0 in ax2 + bx + c, the shape of the parabola is ∪ (opening upwards)
  • If a < 0 in ax2 + bx + c, the shape of the parabola is ∩ (opening downwards).

In the given figure, the graph of a quadratic polynomial x2 - 3x - 4 is shown. It intersects x-axis at (-1, 0) and (4, 0). Therefore, its zeroes are -1 and 4. Here, a > 0, so the graph opens upwards.Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10

Whereas the following figure is a graph of the polynomial - x2 + x + 6. Since it intersects the x-axis at (3, 0) and (-2, 0). Therefore, the zeroes of - x2 + x + 6 are -2 and 3.
Here a < 0, so the parabola opens downwards. 

Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10

Note: In the case of Quadratic polynomial - at most 2 zeroes, Cubic polynomial - at most 3 zeroes, Biquadratic polynomial - at most 4 zeroes.

Question for Important Definitions & Formulas: Polynomials
Try yourself:What is the shape of a parabola when the value of 'a' is negative in the equation ax2 + bx + c?
View Solution

2. Relationship between Zeroes and Coefficients of Polynomials

  • For a quadratic polynomial,
    p(x) = ax2 + bx + c,
    Sum of Roots: α + β = -b/a
    Product of Roots: αβ = c/a 
    where α and β are the zeroes of the polynomial p(x) = ax2 + bx + c
  • For a cubic polynomial,
    p(x) = ax3 + bx2 + cx + d,
    Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10
    where α, β and γ are the zeroes of the polynomial p(x) = ax3 + bx2 + cx + d.
Question for Important Definitions & Formulas: Polynomials
Try yourself:What is the quadratic polynomial whose sum and the product of zeroes is √2, ⅓ respectively?
View Solution

3. Division Algorithm (Deleted from NCERT Textbook)

If we divide a polynomial p(x) by a polynomial g(x), then there exists polynomials q(x) and r(x) such that, p(x) = g(x) x q(x) + r(x), where r(x) = 0 or deg r(x) < deg g(x).

To divide one polynomial by another, follow the steps given below.
Step 1: Arrange the terms of the dividend and the divisor in the decreasing order of their degrees.
Step 2: To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor  Then carry out the division process.
Step 3: The remainder from the previous division becomes the dividend for the next step. Repeat this process until the degree of the remainder is less than the degree of the divisor.

The document Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
All you need of Class 10 at this link: Class 10
126 videos|457 docs|75 tests

Top Courses for Class 10

FAQs on Important Definitions & Formulas: Polynomials - Mathematics (Maths) Class 10

1. What is the degree of a polynomial?
Ans. The degree of a polynomial is the highest power of the variable in the polynomial expression.
2. How can we find the zeroes of a polynomial?
Ans. The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero. To find the zeroes, we set the polynomial equal to zero and solve for the variable.
3. How is the graph of a quadratic polynomial represented?
Ans. The graph of a quadratic polynomial is a parabola, which can be either concave upwards or concave downwards depending on the leading coefficient of the polynomial.
4. What is the relationship between the zeroes and coefficients of a polynomial?
Ans. The relationship between the zeroes and coefficients of a polynomial is given by Vieta's formulas, which state that the sum of the zeroes is equal to the opposite of the coefficient of the linear term, and the product of the zeroes is equal to the constant term divided by the leading coefficient.
5. Why was the Division Algorithm deleted from the NCERT textbook?
Ans. The Division Algorithm was deleted from the NCERT textbook because it is considered an advanced topic that is not essential for the understanding of polynomials at the Class 10 level.
126 videos|457 docs|75 tests
Download as PDF
Explore Courses for Class 10 exam

Top Courses for Class 10

Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10

,

Objective type Questions

,

Free

,

shortcuts and tricks

,

past year papers

,

Viva Questions

,

Summary

,

Exam

,

Previous Year Questions with Solutions

,

Semester Notes

,

Extra Questions

,

Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10

,

Important Definitions & Formulas: Polynomials | Mathematics (Maths) Class 10

,

mock tests for examination

,

Sample Paper

,

MCQs

,

video lectures

,

pdf

,

Important questions

,

ppt

,

study material

,

practice quizzes

;