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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}
Two polynomial are said to be equal if and only if
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.
Here,
Given expressions:3x^{3}+4
This is the polynomial because in3x^{3}, exponent of x=3 is a whole number.
Here,
Given expressions\(\frac{7}{3}\)x^{4}is a polynomials because exponent of x=4 is a whole number.
Here,
Given expressions\(\frac{7}{3}\)x^{4}is a polynomials because exponent of x=4 is a whole number.
Here,
Given expressions:5
Here,5=5+0.x+0.x^{2}.+......+0.x^{4} can be written.So it is a polynomial.
Here,
Given expression:√5.
√5=√5+0.x+0.x^{2}+......+0.x^{4}can be written.So it is polnomial.
Here,
Given expressions =-0.6x
Here numerical coefficient of x =-0.6.Ans.
Here,given ,x in 3xy.
here,3xy=(3y)x
∴Lateral coefficient of x=y.Ans.
Here,
Given ,5xy=(5x)y
∴ Lateral coefficient of y^{2}=x.Ans.
Here,
given, 5 xy=(5x)y
∴coefficient of y =5x.Ans.
Here given,\(\frac{xy}{10}\)=(\(\frac{x}{10}\))y
∴ coefficient of y=\(\frac{x}{10}\).Ans.
Here given, monomial=4xy
Here,Sum of exponents of x and y (1+1)=2
∴degree of 4xy=2.Ans.
Here given, monomial =xyz
Here, Sum of exponents of x,y and z=1+1+1=3
∴degree of xyz=3.Ans.
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.
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.
Here given polynomials√6x +y^{5}
Here, exponent of terms y^{5} is in 5.
∴degree of the polynomials =5. Ans.
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.
Here, given,4y^{3}+2y^{2}+7
Arranging in descending order we get,4y^{3}+2y^{2}+7.Ans.
Find the sum of the following polynomials and also find the degree of the sum:
p(x) =x^{3}- 5x^{2}+ x + 2 and g(x) = x^{3}-3x^{2}+2x+1
1x^{3}-8x^{2}+3x+3;4
2x^{3}-8x^{2}+3x+3;3
2x^{3}-5x^{2}+3x+3;3
2x^{3}-6x^{2}+3x+3;3
Find the sum of the following polynomials and also find the degree of the sum:
p(x) =3x^{2 }+ 5x -2 and g(x) = -3x^{2}-5x + 6
3;0
2;1
4;0
4;1
Find the sum of the following polynomials and also find the degree of the sum:
p(y) =y^{6}-3y^{4} and g(y)= y^{4}+y^{3}+2y^{2}-6
y^{6}-2y^{4}+y^{3}-2y^{2}-6;6
y^{2}-2y^{4}-y^{3}+2y^{2}-6;6
y^{6}-2y^{4}+y^{3}+2y^{2}-6;6
y^{1}-2y^{4}+y^{3}+1y^{2}-6;6
Subtract the second polynomial from the first and find the degree of the difference:
p(y) =y^{3}-3y^{2}+ y + 2 and g(y) =y^{3 }+ 2y+ 1
3y^{2}-y-1;2
-3y^{2}-y+1;2
-3y^{2}-y-1;2
6y^{2}-y+1;2
Find the product of the polynomials and also find the degree of the product:
p(x) =(x+4) and g(x)=(x+6)
x^{2}+10x+2;2
x^{2-}12x+24;2
x^{2}+10x+24;2
x^{2}+10x-24;2
Find the product of the polynomials and also find the degree of the product:
p(x) = (x^{2}-2x+1) and g(x)= (-1 +x)
x^{3}-5x^{2}+3x-1;3
x^{3}-3x^{2}+3x-1;3
x^{3}-3x^{2}-3x-2;3
x^{2}-3x^{2}+3x-1;2
Find the product of the polynomials and also find the degree of the product:
p(x) = (x^{3}+ x + 1) and g(x) = (x-1)
x^{4}+x^{3}+x^{2}-1;2
x^{4}-x^{3}-x^{2}-1;6
x^{4}-x^{3}-x^{2}-1;1
x^{4}-x^{3}+x^{2}-1;4
Find the product of the polynomials and also find the degree of the product:
p(x) = (x^{2}+ x + 1) and g(x) = (x^{2}-x +1)
x^{4}+x^{2}-2;4
x^{4}-x^{2}+1;4
x^{4}+x^{2}+1;4
x^{2}+x^{2}+1;4
Find the product of p(x).q(x) and find the degree of this product:
p(x) = 2x+1 and q(x)= 3x-2
4x^{2}-x-2;2
5x^{2}-x-2;2
6x^{2}+x-2;2
6x^{2}-x-2;2
Find the product of p(x).q(x) and find the degree of this product:
p(x) = x^{2}+ 2x +4 and q(x)= x-2
x^{3}-8;3
x^{3}+8;3
x^{2}-8;3
x^{3}-9;3
Find the value of 'k' .If p(x) =x^{5}+x^{3}(3k-7)x^{2}-2x+6 and q(x)= x^{5}+x^{3}-(1-k)x^{2}-2x+6 are equal.
10
3
6
1
Polynomial h(x) = x^{3}+(a+b)x^{2}-4x+2 and k(x) =x^{3}+5x^{2}-(2a-b)x+2 are equal find the value of a and b.
1,1
3,2
2,3
4,5
Find the sum of the following polynomials and also find the degree of the sum:
p(x) = 3x^{2}+5x-4 and g(x) = 6x^{4}-x + 2
6x^{4}+3x^{2}+4x-2;5
6x^{4}+3x^{2}+8x+2;4
6x^{4}+3x^{2}+4x-2;4
6x^{4}-2x^{2}+4x-2;4
What should be subtracted from the sum ofx^{3}+2x+1 and x^{2}+6x-2 to get 1?
ASK ANY QUESTION ON Polynomials
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