This document summarizes key concepts about polynomials: 1) It defines a polynomial as an algebraic expression involving variables and their powers, and defines the degree of a polynomial as the highest power of the variable. 2) It describes different types of polynomials based on degree, including constant, linear, quadratic, cubic, and biquadratic polynomials. 3) It explains that a zero of a polynomial is a value that makes the polynomial equal to zero when substituted for the variable, and discusses the relationship between zeros and coefficients of polynomials.

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,∫x+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 , ∫3x +5 etc .
• The most general form of a linear
polynomial is ax + b , a ≠ 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 ≠ 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 ≠ 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 ≠ 0 has exactly
one zero .

15. QUADRATIC POLYNOMIAL
• For any quadratic polynomial ax² + bx +c,
a ≠ 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 α and β
are the zeroes of a quadratic polynomial p(x) = ax² + bx +
c , a ≠ 0 , then we know that x − α and x− β are the factors
of p(x) . Therefore ,
• ax² + bx + c = k ( x − α) ( x − β ) ,
• Where k is a constant = k[x² − (α + β)x +αβ]
• = kx² − k( α + β ) x + k αβ
• 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 α , β , Y are the zeroes of a
cubic polynomial ax³ + bx² + cx + d , then
∀ α+β+Y = − b÷a
• = − ( Coefficient of x² ) ÷ coefficient of x³
∀ αβ +βY +Yα =c ÷ a
• = coefficient of x ÷ coefficient of x³
∀ αβY = − 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) ≠ 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