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POLYNOMIALS 1.
Polynomial: Algebraic expressions in which the variables involved with only nonnegative integral exponents are called polynomials.
2.
Terms of a polynomial and their coefficient: If p( x ) = a0 x n + a1 x n −1 + a 2 x n −2 + ... + a n −1 x + a n is a polynomial in variable x, then a0 x n , a1 x n −1,....a n −1 x, a n are known as the terms of polynomial a0, a1, a 2 ,....,a n are known as their coefficients.
p(x )
and
3.
Quadratic polynomial: A polynomial of degree 2 is called quadratic polynomial or any polynomial of the form ax 2 + bx + c where a, b, c are real and a 0 is called quadratic polynomial.
4.
Cubic polynomial: A polynomial of degree 3 is called cubic polynomial. A polynomial of the form ax 3 + bx 2 + cx + d where a, b, c, d are real and a 0 is called cubic polynomial.
5.
Zeroes of a polynomial. k is said to be zero of a polynomial p(x) if p(k) = 0
6.
Graph of polynomial. (i) Graph of a linear polynomial ax + b is a straight line. (ii) A linear polynomial has exactly one zero. (iii) Graph of a quadratic polynomial p( x ) = ax 2 + bx + c is a parabola open upwards like if a 0 . 2 (iv) Graph of a quadratic polynomial p( x ) = ax + bx + c is a parabola open
downwards like if a 0 .
(v) A quadratic polynomial has at most two zeros. (vi) In general a polynomial p(x ) of degree n crosses the x-axis at atmost n points. 7.
Discriminate of a quadratic polynomial. For p( x ) = ax 2 + bx + c, b 2 − 4ac is known as its discriminate ‘D’. D = b 2 − 4ac (i) If D 0 , graph of p( x ) = ax 2 + bx + c will intersect the x-axis at two distinct points, x-coordinates of points of intersection with x-axis are known as ‘zeroes’ of p(x ) .
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x
O
y=ax2+bx+c y
(ii) If D = 0 , graph of p( x ) = ax + bx + c will touch the x-axis at one point only. 2
p(x ) will have only one zero.
y
x
x
O Q y
y=ax2+bx+c
(iii) If D 0 , graph of p( x ) = ax 2 + bx + c will neither touch nor intersect the x-axis. p(x ) will not have any real zeroes. y
x
O Q
x
y
y=ax2+bx+c
8.
Relationship between the zeroes and the coefficients of a Polynomial. (i) For quadratic polynomial (a) If , are zeroes of p( x ) = ax 2 + bx + c , then Sum of zeroes = + =
+=
−b a − (coefficie nt of x )
Coefficien t of x 2 c Product of zeroes = = a (Constant term) = Coefficien t of x 2 For Example: To find the zeros of the quadratic polynomial x 2 − 2 x − 8 and verify the relationship between the zeros and their coefficients. Given quadratic polynomial is
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https://www.entrancei.com/ y = x 2 − 2x − 8 = x 2 − 4 x + 2x − 8 = x( x − 4) + 2( x − 4) = ( x + 2)( x − 4) = ( x + 2)( x − 4) y = 0 gives x = –2, 4. These are two zeros such that = –2, = 4. ( −2 ) b =− Sum of zeros ( + ) = –2 + 4 = 2 = − 1 a c 8 Product of zeros () = – 2 × 4 = – 8 = − = a 1
(b) If , are roots of quadratic polynomial p(x ) , then
p( x ) = k x 2 − ( + )x +
p( x ) = k x − (Sum of zeroes) x + Product of zeroes or p(x) = k[(x – ) (x – )] 2
For Example: To find a quadratic polynomial, the sum and product of whose zeros are
1 and –1 2
respectively. 1 , = –1 2 Required polynomial is f ( x ) = k [ x 2 − ( + )x + ( )]
Here + =
1 = k x 2 − x − 1 where k is a non-zero constant. 2
(ii) For cubic polynomial (a) If , and are zeroes of p( x ) = ax 3 + bx 2 + cx + d − (Coefficie nt of x 2 ) −b ,++= a Coefficien t of x 3 c Sum of product of roots taken two at a time = a Coefficien t of x + + = Coefficien t of x 3
Sum of roots =
Product of roots
= =
−d a − (Constant term) Coefficien t of x 3
For Example: To verify the relationship between the zeros and co-efficient of cubic polynomial x 3 − 4 x 2 + 5 x − 2 whose zeros are 2, 1 and 1. Here, = 2, = 1, = 1 ( −4) b + + = 2 + 1+ 1 = 4 = − =− 1 a d (−2) = − = 2 1 1 = − 1 a
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5 c = 1 a
Hence the result. (b) If , and are zeroes of a cubic polynomial p(x ) , then
p( x ) = k x 3 − ( + + )x 2 + ( + + )x − ( )
x 3 − (Sum of zeroes) x 2 + (Sum of product of zeroes taken two at a time) x p( x ) = k − (Product of zeroes) or p(x) = k[(x – ) (x – ) (x – )]
For Example: To form a cubic polynomial with zeros = 3, = 2, = –1. Here, = 3, = 2, = –1 Required polynomial = k [( x − )( x − )( x − )] = k [( x − 3)( x − 2)( x + 1)] = k [( x 2 − 5 x + 6)( x + 1)] = k [ x 3 − 5 x 2 + 6 x + x 2 − 5 x + 6] = k [ x 3 − 4 x 2 + x + 6] where k is a non-zero constant. 9.
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 ) = q (x ) × g (x ) + r (x ), where r (x ) = 0 or degree of r (x ) < degree of g (x ). Step 1: Divide the highest degree term of the dividend by the highest degree term of the divisor and obtain the remainder. Step 2: If the remainder is zero or degree of remainder is less than divisor, then we cannot continue the division any further. If degree of remainder is equal to or more than divisor repeats step 1. For Example: Apply the division algorithm to find the quotient and remainder on dividing p(x ) by g (x ) as given below. p( x ) = x 3 − 3 x 2 + 5 x − 3 , g ( x ) = x 2 − 2 x–3
x2 − 2
–
x 3 − 3x 2 + 5x − 3 x3 – 2x
+
–3x2 + 7x –3 –3x2 +6
+
q (x ) = x − 3
–
7x – 9 r ( x ) = 7x − 9
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