Important definitions and formulas - Polynomials Class 10 Notes | EduRev

Mathematics (Maths) Class 10

Class 10 : Important definitions and formulas - Polynomials Class 10 Notes | EduRev

The document Important definitions and formulas - Polynomials Class 10 Notes | EduRev is a part of the Class 10 Course Mathematics (Maths) Class 10.
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Polynomial 

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.

Note:

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

1. an ≠ 0

2. a0, a1 x1, a2 x2, a3 x3, ..... an - 1 xn - 1, anxn are terms.

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

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, 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 and formulas - Polynomials Class 10 Notes | EduRev

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.

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.

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 and formulas - Polynomials Class 10 Notes | EduRev

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 and formulas - Polynomials Class 10 Notes | EduRev 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 and formulas - Polynomials Class 10 Notes | EduRev

Note: 

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

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 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 and formulas - Polynomials Class 10 Notes | EduRevWhereas 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 and formulas - Polynomials Class 10 Notes | EduRev

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

Relationship between Zeroes and Coefficients of Polynomials

  • For a quadratic polynomial,
    p(x) = ax2 + bx + c, α + β = -b/a, 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 and formulas - Polynomials Class 10 Notes | EduRev
    where α, β and γ are the zeroes of the polynomial p(x) = ax3 + bx2 + cx + d.
Try yourself:What is the quadratic polynomial whose sum and the product of zeroes is √2, ⅓ respectively?
View Solution

Division Algorithm
If we divide a polynomial p(x) by a polynomial g(x), then their 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.

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