This document discusses key concepts about polynomials including: 1) It defines polynomials as algebraic expressions involving one variable with a finite number of terms. The degree of a polynomial is defined 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 the zeros of a polynomial are the values that make the polynomial equal to zero when substituted for the variable. The zeros are important for understanding the geometric meaning and factorization of polynomials. 4) Relationships are described between the zeros and coefficients of quadratic and cubic polynomials based on their factorizations. The division algorithm for polynomials is also summarized

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