Class 10 – algebra

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Algebra
  • Relation between zeroes and co-efficients of a polynomial
  • Solving simultaneous linear equations – Algebraic method
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Algebra
  • Solving simultaneous linear equations – Graphical method
  • Quadratic equation
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Algebraic expression:Combination of literals and numbers connected by one or more of the four fundamental operations (+, – , ×, ÷ ) is called an algebraic expression.

Example:

Polynomial:Polynomial is a algebraic expression whose literal powers are non-negative integers.

Example:

Polynomial in one variable: A polynomial of the form   

Where are real coefficients and  is called a polynomial in one variable x. It is denoted by p(x), where n is non-negative integer.

Example:

i)                is a polynomial of degree two and is called second degree polynomial (or) quadratic polynomial.

ii)                is a polynomial of degree three and is called third degree polynomial (or) cubic polynomial.

 

iii)            is a polynomial of degree seven, as seven is the highest power of x.

iv)            is not a polynomial since the degree of x is

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i)              Every polynomial is an algebraic expression but every algebraic expression need not be a polynomial.

ii)            The highest power of x is called the degree of the polynomial.

 

Types of polynomials (based on degree):

1.             Linear polynomial: A polynomial of degree one is called a linear polynomial.

Examples:

General Form: , where a and b are real numbers and .

2.             Quadratic polynomial: A polynomial of degree two is called a linear polynomial.

Examples:

General Form: , where a, b

and c are real numbers and .

3.              Cubic polynomial: A polynomial of degree 3 is called a cubic polynomial.

Examples:

General Form: , where a, b, c and d are real numbers and .

4.              Biquadratic polynomial: A polynomial of degree 4 is called a biquadratic polynomial.
They are called quadratic polynomials.

Examples:

General Form: , where a, b, c, d and e are real numbers and .

Value of a polynomial: The value of a polynomial p(x) is obtained on replacing x by k and it is represented by p(k)

Example:
 

Zero of a polynomial: If the value of the polynomial p(x) is 0 at  then ‘k’ is called zero of the polynomial.

Example:Let

Here

 2 is zero of the polynomial .

Method to find zero of a polynomial:

Step 1: Equate polynomial  to zero i.e., .

Step 2: Solve the equation to get the value of the variable.

Step 3: The value obtained will be the zero of the polynomial.

Example:Find the zeroes of the following:

i)          ii)

Solution:

i)

Graphical representation of a polynomial:

i) Linear polynomial:

Let

Case 1: If graph of p(x) is a straight line parallel to x – axis, i.e., it does not meet the axis of x, hence has no zero.

Case 2: If graph of  is a straight line, not parallel to - axis, then it meets the - axis at a point and hence has only one zero.

 A linear polynomial either has no zero means at most one zero.

ii) Quadratic polynomial:

Let

It is represented by a curve called “parabola” opening upward or downward.

If a > 0, parabola opens upward, whereas for a < 0, parabola opens downward.

Case (i):Here, the graph cuts x – axis at two distant points A and B are two zeros of the quadratic polynomial  .

Case (ii):Here, the graph cuts x – axis exactly one point, i.e., at two coincident points. So, the two points  of case (i) coincide here to become one point A.

Case (iii):Here, the graph is either completely above the x – axis or completely below the x – axis. So, it does not cut the x – axis at any point.

So, the quadratic polynomial  has no zero in this case. 

Description: http://upload.wikimedia.org/wikipedia/commons/0/02/Simple_Parabola.svg

iii) Cubic polynomial:

Let

Case (i): If  has 3 zeroes                       Case (ii): If  has 2 zeroes

                     

Case (iii): If  has 1 zero                                   Case (iv): If has no zero

                                  

Description: http://images.clipartpanda.com/bloc-clipart-note-md.png (i) A quadratic polynomial has no zero or one zero or 2 zeroes means at most 2 zeroes.

(ii)   A cubic polynomial has no zero or one zero or 2 zeroes or 3 zeroes means at most 3 zeroes.

Division algorithm:In polynomials, division algorithm is similar to Euclid’s division algorithm, which can be stated as follows:

If  are any two polynomials with , then we can find polynomials  and  such that  where  or degree of  < degree of .

i)               If  is a factor of

ii)             If  then

iii)            If  then

Division of polynomials:

Steps:

i)               Arrange the terms of the dividend and the divisor in descending order of their degrees.

ii)             Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.

iii)            Multiply the whole divisor by the first term of the quotient and subtract the product from the dividend.

iv)            Consider the remainder as the new dividend and proceed as earlier.

v)             Continue this process till a remainder is obtained which is either 0 or whose degree is less than that of the divisor. 

Example: Divide  and verify by division algorithm.

Solution:

     

Here

Verification:

                

                         

Hence proved.

Example: Divide , and verify the division algorithm.

Solution:

     

We stop here since degree of the remainder is less than the degree of  the divisor.

So, quotient , remainder =  = 3.

Now,

                        Dividend = Divisor × Quotient + Remainder

                              

         

                              Hence proved.

Turning points of a polynomial: A turning point of a polynomial is a point where the graph of the polynomial changes from sloping downwards to sloping upwards, or vice versa. So, the gradient changes from negative to positive (or) positive to negative.

i)               A quadratic polynomial has only one turning point.

ii)             A cubic polynomials could have up to two turning points, and so would look something like this.

                            

Description: http://www.aids.harvard.edu/img/news/spotlight/newsletter_layout/logo.gif

i)               A polynomial of degree n have at most n zeroes.

ii)             A polynomial of degree n can have up to (n − 1) turning points.