CBSE Grade 10 Maths Chapter 2 - Polynomials

Introduction to Algebraic Expression

•  An algebraic expression is an expression made up of variables and constants along with mathematical operators.

• A variable is a symbol, usually a letter, which is used to represent an unknown number. 

• A term can be a number, a variable, or a number and variable combined by multiplication or division .In other words , a term is a product of variables and constants. A term can be an algebraic expression in itself.

• An expression can be term or a collection of terms separated by addition or subtraction operators. 

• The number (positive or negative) in the algebraic term is called the coefficients of the algebraic terms . The coefficient of 1 in an algebraic term is usually not written.

• An algebraic expression can have any number of terms. The coefficient in each term can be any real number. There can be any number of variables in an algebraic expression. The exponent on the variables, however, must be rational numbers.
• Example : 
  

 

What are Polynomials?

Polynomial is an algebraic expression in which the exponent on any variable is a whole number. 

Example :

• 2x³+7x-1  is an example of a polynomial. It is an algebraic expression as well.

•  5x⁵+3√x is an algebraic expression, but not a polynomial. – since the exponent on x is 1/2 which is not a whole number. 

Some other examples of polynomials are 10, a + b, 7x + y + 5, w + x + y + z, etc.

Degree of a Polynomial

• For a polynomial in one variable – the highest exponent on the variable in a polynomial is the degree of the polynomial.

Example: 

The degree of the polynomial x²-7x+19 is 2, as the highest power of x in the given expression is x².

The degree of p(x) = x⁵ – x³ + 7 is 5.

Types Of Polynomials

Polynomials can be classified based on:

a) Number of terms

b) Degree of the polynomial.

Types of polynomials based on the number of terms

a) Monomial – A polynomial with just one term. Example: 7x, 23x², 69xy

b) Binomial – A polynomial with two terms. Example: 4x²-3x, 25x+41

a) Trinomial – A polynomial with three terms. Example: x²-3x+94

Types of Polynomials based on Degree

a) Constant Polynomial 

• A polynomial whose degree is zero is called a Constant Polynomial.
  

For example: 7/6 , -16 , 64 , √34 are constant polynomials.

b) Linear Polynomial

• A polynomial whose degree is one is called a linear polynomial.

For example: 12x+1,  1/(2x – 7), √s + 5 are linear polynomials.

c) Quadratic Polynomial

• A polynomial of degree two is called a quadratic polynomial.
• In general, a quadratic polynomial can be expressed in the form ax² + bx + c, where a≠0 and a, b, c are constants.
  

For example: 3x²+8x+5, x² – 9, a² + a + 7 are quadratic polynomials.

d) Cubic Polynomial

• A polynomial of degree three is called a Cubic Polynomial.
• In general, a quadratic polynomial can be expressed in the form ax³ + bx² + cx + d, where a≠0 and a, b, c, d are constants.
  

For example, 3x³+8x+5 , x³ – 9x +2, a³ + a² + √a + 7 are cubic polynomials.

 

 Value of Polynomial

Let p(y) is a polynomial in y and α could be any real number, then the value calculated after putting the value y = α in p(y) is the final value of p(y) at y = α. This shows that p(y) at y = α is represented by p (α).

Zero of a Polynomial

• A zero of a polynomial p(x) is the value of x for which the value of p(x) is 0. 
• If k is a zero of p(x), then p(k)=0.
• In general, if k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e., k = -b/a. Hence, the zero of the linear polynomial ax + b is –b/a = -(Constant term)/(coefficient of x)
  

For example, consider a polynomial p(x) = 3x - 6.

When x=2, the value of p(x) will be equal to
p(2)=3×2-6
=6-6
=0

Since p(x)=0 at x=2, we say that 2 is a zero of the polynomial 3x - 6.

• Graphically, Zeros of the polynomials are the x coordinates of the point where the graph of that polynomial intersects the x-axis.

Number of Zeros

In general, a polynomial of degree n has at most n zeros.

1. A linear polynomial has one zero,
2. A quadratic polynomial has at most two zeros.
3. A cubic polynomial has at most 3 zeros.

 Example : Find the number of zeros of the following curves.

Graphical Representations of Equations on a Graph

• Any equation can be represented as a graph on the Cartesian plane, where each point on the graph represents the x and y coordinates of the point that satisfies the equation.
•  An equation can be seen as a constraint placed on the x and y coordinates of a point, and any point that satisfies that constraint will lie on the curve

Geometrical Representation of a Linear Polynomial

• For a linear polynomial ax + b, a ≠ 0, the graph of y = ax + b is a straight line which intersects the x-axis at exactly one point, namely, (-b/a , 0) .
  

• Therefore, the linear polynomial ax + b, a ≠ 0, has exactly one zero, namely, the x-coordinate of the point where the graph of y = ax + b intersects the x-axis.

 Examples of linear polynomials are shown in the graphs given below.

 Geometrical Representation of a Quadratic Polynomial

• The graph of a quadratic polynomial is a parabola
• It looks like a U which either opens upwards or opens downwards depending on the value of ‘a’ in ax²+bx+c
• If ‘a’ is positive, then parabola opens upwards and if ‘a’ is negative then it opens downwards
• It can cut the x-axis at 0, 1 or two points
  

 

 Graph of the polynomial y = xⁿ

For a polynomial of the form y = xⁿ where n is a whole number:

• as n increases, the graph becomes steeper or draws closer to the Y-axis
• If n is odd, the graph lies in the first and third quadrants
• If n is even, the graph lies in the first and second quadrants
• The graph of y=−xⁿ is the reflection of the graph of y=xⁿ on the x-axis

 

Relationship between Zeroes and Coefficients of a Polynomial

For Quadratic Polynomial: 

• If α and β are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, a ≠ 0, then we know that (x – α) and (x – β) are the factors of p(x). 

• α +  β = -b/a

  Sum of zeroes = -coefficient of x /coefficient of x²

•  αβ = c/a

  Product of zeroes = constant term / coefficient of x²

For Cubic Polynomial

If α,β and γ are the roots of a cubic polynomial ax³+bx²+cx+d, then

α+β+γ = -b/a

αβ +βγ +γα = c/a

αβγ = -d/a

 Division Algorithm for Polynomials

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of 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.

 One more Example :

Algebraic Identities

1. (a+b)²=a²+2ab+b²
2. (a−b)²=a²−2ab+b²
3. (x+a)(x+b)=x²+(a+b)x+ab
4. a²−b²=(a+b)(a−b)
5. a³−b³=(a−b)(a²+ab+b²)
6. a³+b³=(a+b)(a²−ab+b²)
7. (a+b)³=a³+3a²b+3ab²+b³
8. (a−b)³=a³−3a²b+3ab²−b³