Subject: Optional Mathematics

The collection of numbers and alphabets linked by the sign of multiplication and division is called algebraic term. 3, x, 5x, 9xy, \(\frac{2x^2}{4y^2}\), etc are some examples of an algebraic term. The term with no variable is called constant term. The number or alphabets by which variable is multiplied is called the coefficient of the term. 3 is a coefficient of 3x. The number used for the coefficient is called numerical coefficient and alphabet used for coefficients are called literal coefficients. In the term 5ax,5 is numerical coefficient, 'a' is a literal coefficient and 'x' is avariable. In 7x^{3} , 7 is coefficient, x is base and 3 is power.

f(x) =a_{n}x^{n} + a_{n-1} x^{n-1} + .........................+ a_{o}or. f(x) = a_{o} + a_{1}x+a_{2}x + ............................ + a_{n}x^{n}

- If a
_{n}≠ 0, then n is called the degree of the polynomial f(x). The degree of the polynomial can never be negative. - a
_{n}x^{n}, a_{n-1 }x^{n-1},....................... a_{o}are called the terms of the polynomial f(x). a is called the constant term. - a
_{o}a_{1},a_{2}, ....... and are called the coefficients of the polynomial f(x). - If all a
_{n }≠ 0, then a_{n}x_{n}is called the leading term and is called the leading coefficient of the polynomial. - If all the coefficient a
_{n}, a_{1}, ..... a are zero, then f(x) is called zero polynomial. The degree of zero polynomial is never defined. - The degree of a polynomial is zero if and only if it is a non-zero constant polynomial. f(x) =5 is a zero degree polynomial.

Two polynomial are said to be equal if and only if

- Their degrees are same
- Coefficients of corresponding terms are same.

If f(x) = a_{n}x^{n} + a_{n-1}x^{n-1}+ .............a_{o} and g(x) = bmx^{m}_{+}bm^{-1}x^{m-1} + ....................+ b_{o} are equal polynomials, then n=m,a_{n} = bm.

a_{n-1} = b_{m-1}, ......a_{o} = b_{o.}If polynomials f(x) and g(x) are equal, then we write f(x) =g(x).

Example :

If f(x) = 3x^{2}+2x+1 and g(x) = \(\frac{6}{2}\)x^{2}+\(\frac{4}{2}\)x+ \(\frac{2}{2}\) then f(x) = g(x)

Let f(x)and g(x) be two polynomials given as follows:

f(x) = a_{0} +a_{1}x+a_{2} x^{2} +.....+ a_{n}x^{n}g(x) =b_{0}+ b_{1}x+b_{2}x^{2}+..........+b_{n}x^{n}

Then, their sum is defined as

f(x) = (a_{0}+b_{0}) + (a_{1}+b_{1})x + (a_{2}+b_{2})x^{2}+ ........+(a_{n}+b_{n})x^{n}.

Thus, the sum of two polynomials can be found by grouping the power terms, retaining their signs and adding the coefficients of like power.

For the calculation of sum or difference of polynomials, the polynomials are first kept in standard form. Then, the coefficients of like terms are added for sum and subtracted for the difference.

**Closure property**If the sum of two polynomials over a set is again a polynomial over the same set, then polynomials are said to be closed under addition.

The sum of two polynomials over real numbers is also a polynomial over real numbers are closed under addition.

It means, the sum of two polynomials is again a polynomial.**Commutative property**

For ant two polynomials f(x) and g(x),

f(x) + g(x) = g(x)+f(x)

This property is called the commutative property of addition.**Associative property**

For any three polynomials f(x), g(x) and h(x) f(x) + {g(x) + h(x)} = {f(x) + g(x)} + h(x).

This property is called the associative property of addition.**Additive identity**

For any polynomial f(x), there is a polynomial 0 such that.

f(x)+0 = f(x)

Then 0 is called additive identity.**Additive Inverse**

For any polynomial f(x) there is a polynomial - f(x) such that

f(x) + {-f(x)} =0, the additive identity. Then -f(x) is called additive inverse of f(x).

- If the divisor is x+2, then a = -2
- If the divisor is 2x-3, then a =\(\frac{3}{2}\)
- If the divisor is 2x+3, then a = \(\frac{-3}{2}\)
- If a polynomial p(x) is divided by x-a, then the remainder is p(a).

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Which of the following expression are polynomials and which are not polynomials?Give reasons.

3x^{3}+4

Here,

Given expressions:3x^{3}+4

This is the polynomial because in3x^{3}, exponent of x=3 is a whole number.

Which of the following expression are polynomials and which are not polynomials?Give reasons.7

\(\frac{7}{3}\)x^{4}

Here,

Given expressions\(\frac{7}{3}\)x^{4}is a polynomials because exponent of x=4 is a whole number.

Which of the following expression are polynomials and which are not polynomials?Give reasons.

\(\frac{7}{3}\)x^{4}

Here,

Given expressions\(\frac{7}{3}\)x^{4}is a polynomials because exponent of x=4 is a whole number.

Which of the following expression are polynomials and which are not polynomials?Give reasons.

5

Here,

Given expressions:5

Here,5=5+0.x+0.x^{2}.+......+0.x^{4} can be written.So it is a polynomial.

Which of the following expression are polynomials and which are not polynomials?Give reasons.

\(\sqrt{5}\)

Here,

Given expression:√5.

√5=√5+0.x+0.x^{2}+......+0.x^{4}can be written.So it is polnomial.

Find the numberical coefficients of the following expressions:

-0.6x

Here,

Given expressions =-0.6x

Here numerical coefficient of x =-0.6.Ans.

Find the lateral coefficient of:

x in 3xy

Here,given ,x in 3xy.

here,3xy=(3y)x

∴Lateral coefficient of x=y.Ans.

Find the lateral coefficient of:

y^{2} in 3xy^{2}

Here,

Given ,5xy=(5x)y

∴ Lateral coefficient of y^{2}=x.Ans.

Write down the coefficient of y in the following monomials:

5xy

Here,

given, 5 xy=(5x)y

∴coefficient of y =5x.Ans.

Write down the coefficient of y in the following monomials:

\(\frac{xy}{10}\)

Here given,\(\frac{xy}{10}\)=(\(\frac{x}{10}\))y

∴ coefficient of y=\(\frac{x}{10}\).Ans.

Find the degreed of the following monomials:

4xy

Here given, monomial=4xy

Here,Sum of exponents of x and y (1+1)=2

∴degree of 4xy=2.Ans.

Find the degreed of the following monomials:

xyz

Here given, monomial =xyz

Here, Sum of exponents of x,y and z=1+1+1=3

∴degree of xyz=3.Ans.

Find the degreed of the following monomials:

\(\frac{x^2y^2z}{7}\)

Here given, monomial=\(\frac{x^2y^2z}{7}\)

Here,Sum of exponents 2 of x , 2 of y and 1 of z=2+2+1=5

∴degree of\(\frac{x^2y^2z}{7}\)=5.Ans.

Find the degreed of the following monomials:

2x^{2}+8x-9

Here given polynomial2x^{2}+8x-9

Here,the largest exponent of x=2.so,2x^{2}+8x-9,is a second degree polynomial in x.

Find the degreed of the following monomials:

√6x +y^{5}

Here given polynomials√6x +y^{5}

Here, exponent of terms y^{5} is in 5.

∴degree of the polynomials =5. Ans.

Find out the degree of the following polynomials and arrange their terms in ascending order.

3x^{2}+4x-18

Here given polynomials,3x^{2}+4x-18

The largest exponent of x =2.So, it is the second degree polynomial

Arranging in ascending order we get, -18+4x+3x^{2}.Ans.

Arrange the polynomials in descending order of power:

4y^{3}+2y^{2}+7

Here, given,4y^{3}+2y^{2}+7

Arranging in descending order we get,4y^{3}+2y^{2}+7.Ans.

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