Shubhanshu math project work , polynomial

polynomial

1. MATH PROJECT WORK
NAME - SHUBHANSHU BHARGAVA

CLASS -10
SECTION - A
SHIFT- I SHIFT

2. POLYNOMIALS
•POLYNOMIAL – A polynomial in one
variable X is an algebraic expression in
X of the form
NOT A POLYNOMIAL – The
expression like 1¸x - 1,òx+2 etc are not
polynomials .

3. DEGREE OF POLYNOMIAL
•Degree of polynomial- The highest
power of x in p(x) is called the degree of
the polynomial p(x).
•EXAMPLE –
•1) F(x) = 3x +½ is a polynomial in the
variable x of degree 1.
•2) g(y) = 2y² - ⅜ y +7 is a polynomial in
the variable y of degree 2 .

4. TYPES OF POLYNOMIALS
•Types of polynomials are –
•1] Constant polynomial
•2] Linear polynomial
•3] Quadratic polynomial
•4] Cubic polynomial
•5] Bi-quadratic polynomial

5. CONSTANT POLYNOMIAL
•CONSTANT POLYNOMIAL – A
polynomial of degree zero is called a
constant polynomial.
•EXAMPLE - F(x) = 7 etc .
•It is also called zero polynomial.
•The degree of the zero polynomial is not
defined .

6. LINEAR POLYNOMIAL
•LINEAR POLYNOMIAL – A
polynomial of degree 1 is called a linear
polynomial .
•EXAMPLE- 2x-3 , ò3x +5 etc .
•The most general form of a linear
polynomial is ax + b , a ¹ 0 ,a & b are
real.

7. QUADRATIC POLYNOMIAL
•QUADRATIC POLYNOMIAL – A
polynomial of degree 2 is called quadratic
polynomial .
•EXAMPLE – 2x² + 3x - ⅔ , y² - 2 etc .
More generally , any quadratic polynomial
in x with real coefficient is of the form ax² +
bx + c , where a, b ,c, are real numbers
and a ¹ 0

8. CUBIC POLYNOMIALS
•CUBIC POLYNOMIAL – A
polynomial of degree 3 is called a cubic
polynomial .
•EXAMPLE = 2 - x³ , x³, etc .
•The most general form of a cubic
polynomial with coefficients as real
numbers is ax³ + bx² + cx + d , a ,b ,c ,d
are reals .

9. BI QUADRATIC POLYNMIAL
•BI – QUADRATIC POLYNOMIAL –
A fourth degree polynomial is called a
biquadratic polynomial .

10. VALUE OF POLYNOMIAL
•If p(x) is a polynomial in x, and if k is any real
constant, then the real number obtained by
replacing x by k in p(x), is called the value of
p(x) at k, and is denoted by p(k) . For
example , consider the polynomial p(x) = x²
-3x -4 . Then, putting x= 2 in the polynomial ,
we get p(2) = 2² - 3 ´ 2 - 4 = - 4 . The value
- 6 obtained by replacing x by 2 in x² - 3x - 4
at x = 2 . Similarly , p(0) is the value of p(x) at
x = 0 , which is - 4 .

11. ZERO OF A POLYNOMIAL
•A real number k is said to a zero of a
polynomial p(x), if said to be a zero of a
polynomial p(x), if p(k) = 0 . For example,
consider the polynomial p(x) = x³ - 3x - 4 .
Then,
• p(-1) = (-1)² - (3(-1) - 4 = 0
• Also, p(4) = (4)² - (3 ´4) - 4 = 0
• Here, - 1 and 4 are called the zeroes of the
quadratic polynomial x² - 3x - 4 .

12. HOW TO FIND THE ZERO OF
A LINEAR POLYNOMIAL
•In general, if k is a zero of p(x) = ax + b,
then p(k) = ak + b = 0, k = - b ¸ a . So,
the zero of a linear polynomial ax + b is
- b ¸ a = - ( constant term ) ¸
coefficient of x . Thus, the zero of a
linear polynomial is related to its
coefficients .

13. GEOMETRICAL MEANING OF
THE ZEROES OF A POLYNOMIAL
•We know that a real number k is a zero
of the polynomial p(x) if p(K) = 0 . But to
understand the importance of finding
the zeroes of a polynomial, first we shall
see the geometrical meaning of –
•1) Linear polynomial .
•2) Quadratic polynomial
•3) Cubic polynomial

14. GEOMETRICAL MEANING OF
LINEAR POLYNOMIAL
•For a linear polynomial ax + b , a ¹ 0,
the graph of y = ax +b is a straight line .
Which intersect the x axis and which
intersect the x axis exactly one point (-
b ¸ 2 , 0 ) . Therefore the linear
polynomial ax + b , a ¹ 0 has exactly
one zero .

15. QUADRATIC POLYNOMIAL
•For any quadratic polynomial ax² + bx +c,
a ¹ 0, the graph of the corresponding
equation y = ax² + bx + c has one of the
two shapes either open upwards or open
downward depending on whether a>0 or
a<0 .these curves are called parabolas .

16. GEOMETRICAL MEANING OF
CUBIC POLYNOMIAL
•The zeroes of a cubic polynomial p(x) are
the x coordinates of the points where the
graph of y = p(x) intersect the x – axis .
Also , there are at most 3 zeroes for the
cubic polynomials . In fact, any polynomial
of degree 3 can have at most three
zeroes .

17. RELATIONSHIP BETWEEN
ZEROES OF A POLYNOMIAL
For a quadratic polynomial – In general, if a and b
are the zeroes of a quadratic polynomial p(x) = ax² + bx +
c , a ¹ 0 , then we know that x - a and x- b are the factors
of p(x) . Therefore ,
•ax² + bx + c = k ( x - a) ( x - b ) ,
•Where k is a constant = k[x² - (a + b)x +ab]
•= kx² - k( a + b ) x + k ab
• Comparing the coefficients of x² , x and constant term on
both the sides .
•Therefore , sum of zeroes = - b ¸ a
•= - (coefficients of x) ¸ coefficient of x²
•Product of zeroes = c ¸ a = constant term ¸ coefficient of x²

18. RELATIONSHIP BETWEEN ZERO
AND COEFFICIENT OF A CUBIC
POLYNOMIAL
•In general, if a , b , Y are the zeroes of a
cubic polynomial ax³ + bx² + cx + d , then
"a+b+Y = - b¸a
•= - ( Coefficient of x² ) ¸ coefficient of x³
"ab +bY +Ya =c ¸ a
•= coefficient of x ¸ coefficient of x³
"abY = - d ¸ a
•= - constant term ¸ coefficient of x³

19. DIVISION ALGORITHEM FOR
POLYNOMIALS
•If p(x) and g(x) are any two polynomials
with g(x) ¹ 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) .
•This result is taken as division algorithm
for polynomials .

20. THE
END