aman yadav
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Maths Project
• Made By :- Aman Yadav• Class:-Ix-A• Roll no. :- 05
• Submitted To:- Mr. Manish Ahuja (Maths Teacher)• By The Help oF Mrs.Neetu
Sharma (Computer Teacher)
Polynomials
Topics Covered
• Polynomials.
• Exponents And Terms.
• Degree Of A Polynomial In One Variable.
• Degree Of A Polynomial In Two Variables.
• Remainder Theorem.
• Factor Theorem.
• Algebric Identities.
Polynomials
• A polynomial is a monomial or a sum of monomials.
• Each monomial in a polynomial is a term of the polynomial.
The number factor of a term is called the coefficient.
The coefficient of the first term in a polynomial is the lead coefficient.
• A polynomial with two terms is called a binomial. • A polynomial with three term is called a trinomial.
Exponents And Terms.
Degree of a Polynomial in one variable:-
Degree of a Polynomial in one variable. What is degree of the following binomial? The answer is 2. 5x2 + 3 is a polynomial in x of degree 2. In case of a polynomial in one variable, the highest power of the variable is called the degree of polynomial .
Degree of a Polynomial in two variables.
• What is degree of the following polynomial?
49375 332 yxyxyx
In case of polynomials on more than one variable, the sum of powers of the variables in each term is taken up and the highest sum so obtained is called the degree of polynomial.
• The answer is five because if we add 2 and 3 , the answer is five which is the highest power in the whole polynomial.
E.g.- is a polynomial in x
and y of degree 7.
92853 243 yxyxyx
Polynomials in one variable
The degree of a polynomial in one variable is the largest exponent of that variable.
14 x
A constant has no variable. It is a 0 degree polynomial.2This is a 1st degree polynomial. 1st degree polynomials are linear.
1425 2 xx This is a 2nd degree polynomial. 2nd degree polynomials are quadratic.
183 3 x This is a 3rd degree polynomial. 3rd degree polynomials are cubic.
Examples
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Polynomials Degree Classify by degree
Classify by no. of terms.
5 0 Constant Monomial
2x - 4 1 Linear Binomial
3x2 + x 2 Quadratic Binomial
x3 - 4x2 + 1 3 Cubic Trinomial
Remainder Theorem
TEXT TEXT TEXT TEXT
Let f(x) be a polynomial of degree n > 1 and let a be any real number.
When f(x) is divided by (x-a) , then the remainder is f(a).
PROOF Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder
is r(x).
Then, degree r(x) < degree (x-a)
degree r(x) < 1 [ therefore, degree (x-a)=1]
degree r(x) = 0
r(x) is constant, equal to r (say)
Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r.
Therefore, f(x) = (x-a)*g(x) + r (i)
Putting x=a in (i), we get r = f(a)
Thus, when f(x) is divided by (x-a), then the remainder is f(a).
Questions on Remainder Theorem
Q.) Find the remainder when the polynomial
f(x) = x4 + 2x3 – 3x2 + x – 1 is divided by (x-2).
A.) x-2 = 0 x=2
By remainder theorem, we know that when f(x) is divided by (x-2), the remainder is x(2).
Now, f(2) = (24 + 2*23 – 3*22 + 2-1)
= (16 + 16 – 12 + 2 – 1) = 21.
Hence, the required remainder is 21.
Factor Theorem
Let f(x) be a polynomial of degree n > 1 and let a be
any real number.
(i) If f(a) = 0 then (x-a) is a factor of f(x).
PROOF let f(a) = 0
On dividing f(x) by 9x-a), let g(x) be the quotient. Also, by remainder theorem, when f(x) is divided by (x-a), then the remainder is f(a).
therefore f(x) = (x-a)*g(x) + f(a)
f(x) = (x-a)*g(x) [therefore f(a)=0(given]
(x-a) is a factor of f(x).
Algebraic Identities
Some common identities used to factorize polynomials
(x+a)(x+b)=x2+(a+b)x+ab(a+b)2=a2+b2+2ab (a-b)2=a2+b2-2ab a2-b2=(a+b)(a-b)
Algebraic Identities
Advanced identities used to factorize polynomials
(x+y+z)2=x2+y2+z2
+2xy+2yz+2zx
(x-y)3=x3-y3-3xy(x-y)
(x+y)3=x3+y3+3xy(x+y)
x3+y3=(x+y) * (x2+y2-xy) x3-y3=(x+y) *
(x2+y2+xy)
Q/A on Polynomials
Q.1) Show that (x-3) is a factor of polynomial
f(x)=x3+x2-17x+15.
A.1) By factor theorem, (x-3) will be a factor of f(x) if f(3)=0.
Now, f(x)=x3+x2-17x+15
f(3)=(33+32-17*3+15)=(27+9-51+15)=0
(x-3) is a factor of f(x).
Hence, (x-3) is a factor of the given polynomial f(x).
Q/A on Polynomials
Q.1) Factorize:
(i) 9x2 – 16y2 (ii)x3-x
A.1)(i) (9x2 – 16y2) = (3x)2 – (4y)2
= (3x + 4y)(3x – 4y)
therefore, (9x2-16y2) = (3x + 4y)(3x – 4y)
(ii) (x3-x) = x(x2-1)
= x(x+1)(x-1)
therefore, (x3-x) = x(x + 1)(x-1)
Points to Remember
• A real number ‘a’ is a zero of a polynomial p(x) if p(a)=0. In this case, a is also called a root of the equation p(x)=0.
• Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.
THANK YOUTHANK YOU
With Regards:- Aman Yadav