Important Definitions & Formulas: Polynomials

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

 Table of contents Important Definitions 1. Polynomials 2. Degree of a Polynomial 3. Zeroes of a Polynomial Important Methods and Formulas 1. Graph of a Quadratic Polynomial 2. Relationship between Zeroes and Coefficients of Polynomials 3. Division Algorithm (Deleted from NCERT Textbook)

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.

## 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.

Polynomial

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.

Example:
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:

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

### 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 or

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,  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.

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?

## 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.

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.

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?

## 2. Relationship between Zeroes and Coefficients of Polynomials

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,

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?

## 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.
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## 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.

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