POLYNOMIALS

Chapter -2

POLYNOMISALS

The Polynomial p(x)

   For example, 4x + 2 is a polynomial in the variable x of degree 1, 2y² – 2y + 4 is a polynomial in the variable y of degree
 is a polynomial in the variable x of degree 3 and  is a polynomial in the variable u of  degree 6. Expressions like , are not polynomials.

A polynomial of degree 1 is called a linear polynomial. For example,   are all linear polynomials. Polynomials such as 2x + 5 – x², x³ + 1, etc., are not linear polynomials.

A polynomial of degree 2 is called a quadratic polynomial. The name ‘quadratic’ has been derived from the word ‘quadratie’ which means ‘square’.

 are some examples of quadratic polynomials (whose coefficient are real numbers). More generally, any quadratic polynomial in x is of the form ax² + bx + c, where a, b, c are real numbers and a ≠0.  A polynomial of degree 3 is called a cubic polynomial. Some examples of a cubic polynomial are  In fact, the most general form of a cubic polynomial is ax³ + bx² + cx + d.

Where, a, b, c, d are real numbers and a≠0.
Now consider the polynomial p(x) = x² – 3x – 4. Then, putting x = 2 in the polynomial, we get p(2) = 2² – 3 × 2 – 4 = – 6. The value ‘– 6’, obtained by replacing x by 2 in x² – 3x – 4, is the value of x² – 3x – 4 at x = 2. Similarly, p(0) is the value of p(x) at x = 0, which is – 4.

 If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k). What is the value of p(x) = x² –3x – 4 at x = –1? We have:

p(–1) = (–1)² –{3 × (–1)} – 4 = 0
Also, note that p(4) = 4² – (3 × 4) – 4 = 0.
As p(–1) = 0 and p(4) = 0, –1 and 4 are called the zeroes of the quadratic polynomial  x² – 3x – 4. More generally, a real number k is said to be a zero of a polynomial p(x),  if p(k) = 0.

You have already studied in Class IX, how to find the zeroes of a linear polynomial. For example, if k is a zero of p(x) = 2x + 3, then p(k) = 0 gives us 2k + 3 = 0, i.e.,

 
In general, if k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e.,  

So, the zero of the linear polynomial ax + b is -ba= -Constant term coefficient of ax + b = 

Thus, the zero of a linear polynomial is related to its coefficients. Does this happen in the case of other polynomials too?  For example, are the zeroes of a quadratic polynomial also related to its coefficients?
In this chapter, we will try to answer these questions. We will also study the division algorithm for polynomials.

Geometrical Meaning of the Zeroes of a Polynomial
You know that a real number k is a zero of the polynomial p(x) if p(k) = 0. But why are the zeroes of a polynomial so important? To answer this, first we will see the geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes.

Consider first a linear polynomial ax + b, a ≠ 0. You have studied in Class IX that the graph of y = ax + b is a straight line. For example, the graph of y = 2x + 3 is a straight line passing through the points (– 2, –1) and (2, 7).

From Fig. 2.1, you can see that the graph of y = 2x + 3 intersects the x -axis mid-way between x = –1 and x = – 2, that is, at the point  
You also know that the zero of 2x + 3 is   Thus, the zero of the polynomial 2x + 3 is the x-coordinate of the point where the graph of y = 2x + 3 intersects the x-axis. In general, for a 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 axis.ax+b, a≠0,  The graph of y = ax + b is a straight line which Intersects the x-axis at exactly one point, namely,  Therefore, the linear polynomial

 

Relationship between Zeroes and Coefficients of a Polynomial

 You have already seen that zero of a linear polynomial ax + b is

 We will now try to answer the question raised in Section 2.1 regarding the relationship between zeroes and coefficients of a quadratic polynomial. For this, let us take a quadratic polynomial, say p(x) = 2x² – 8x + 6. In Class IX, you have learnt how to factorise quadratic polynomials by splitting the middle term. So, here we need to split the middle term ‘– 8x’ as a sum of two terms, whose product is 6 × 2x² = 12x². So, we write

2x² – 8x + 6 = 2x² – 6x – 2x + 6 = 2x(x – 3) – 2(x – 3)

                    = (2x – 2)(x – 3) = 2(x – 1)(x – 3)

So, the value of p(x) = 2x² – 8x + 6 is zero when x – 1 = 0 or x – 3 = 0, i.e., when x = 1 or x = 3. So, the zeroes of 2x² – 8x + 6 are 1 and 3. Observe that:
 

 
Let us take one more quadratic polynomial, say, p(x) = 3x² + 5x – 2. By the method of splitting the middle term,

    
 Hence, the value of 3x² + 5x – 2 is zero when either 3x – 1 = 0 or x + 2 = 0, i.e.,

when x=1/3 or x =-2.   So, the zeroes of  Observe that:

 

In general, if α* and β* are the zeroes of the quadratic polynomial p(x) = ax² + bx + c,

a≠0,  then you know that x-α and x-β     are the factors of p(x). Therefore,

 where k is a constant

 

                             

Comparing the coefficients of x², x and constant terms on both the sides, we get
   

 
                          

 

Division algorithm for Polynomials

 You know that a cubic polynomial has at most three zeroes. However, if you are given only one zero, can you find the other two? For this, let us consider the cubic polynomial

x3-3x2-x+3.  If we tell you that one of its zeroes is 1, then you know that x – 1 is a factor of x³ – 3x² – x + 3. So, you can divide x³ – 3x² – x + 3 by x – 1, as you have learnt in Class IX, to get quotient x² – 2x – 3.

Next, you could get the factors of x² – 2x – 3, by splitting the middle term, as (x + 1) (x – 3). This word give you

 

    

So, all the three zeroes of the cubic polynomial are now known to you as 1, – 1, 3.
Let us discuss the method of dividing one polynomial by another in some detail.
​​​​​​​Before noting the steps formally.

 then we can find polynomials q(x) and r(x) such that
 
Where r(x) = 0 or degree of r (x) < degree of g(x).