Answer
Verified
396k+ views
Hint: The degree of a polynomial tells about the number of solutions that function can have and the number of times the function will cross the x-axis. If the degree is zero then then the equation cannot have any solution.
In this question three functions are given whose degree is to be found, here we first find out the product of the function and the highest degree is determined.
Complete step by step solution:
\[p\left( u \right) = {u^2} + 3u + 4\]
\[q\left( u \right) = {u^2} + u - 12\]
\[r\left( u \right) = u - 2\]
All the given equation has the variable u hence we have to find the monomial that has the highest power.
Degree is to be find for the \[p\left( u \right)q\left( u \right)r\left( u \right)\], hence we first find the function whose degree will be find
\[
p\left( u \right)q\left( u \right)r\left( u \right) = \left( {{u^2} + 3u + 4} \right)\left( {{u^2} + u - 12} \right)\left( {u - 2} \right) \\
= \left( {{u^4} + {u^3} - 12{u^2} + 3{u^3} + 3{u^2} - 36u + 4{u^2} + 4u - 48} \right)\left( {u - 2} \right) \\
= \left( {{u^4} + 4{u^3} - 5{u^2} - 32u - 48} \right)\left( {u - 2} \right){\text{ }}\left[ {\because {a^m} \times {a^n} = {a^{m + n}}} \right] \\
= \left( {{u^5} - 2{u^4} + 4{u^4} - 8{u^3} - 5{u^3} + 10{u^2} - 32{u^2} + 64u - 48u + 96} \right) \\
= \left( {{u^5} + 2{u^4} - 13{u^3} - 22{u^2} + 16u + 96} \right) \\
\]
So, the value of \[p\left( u \right)q\left( u \right)r\left( u \right) = \left( {{u^5} + 2{u^4} - 13{u^3} - 22{u^2} + 16u + 96} \right)\]
As we know the degree of a polynomial is the highest power of the nonzero coefficient monomial, so in the obtained polynomial we can see \[{u^5}\]has the highest degree with the coefficient 1 which is the order of the polynomial.
Hence, we can say the degree of \[p\left( u \right)q\left( u \right)r\left( u \right)\] is equal to 5.
Note: When two powers are multiplied together with the same base then their exponents adds up \[{a^m} \times {a^n} = {a^{m + n}}\] here base is \[a\] and the powers are m, n. Two powers with different bases cannot be added together.
In this question three functions are given whose degree is to be found, here we first find out the product of the function and the highest degree is determined.
Complete step by step solution:
\[p\left( u \right) = {u^2} + 3u + 4\]
\[q\left( u \right) = {u^2} + u - 12\]
\[r\left( u \right) = u - 2\]
All the given equation has the variable u hence we have to find the monomial that has the highest power.
Degree is to be find for the \[p\left( u \right)q\left( u \right)r\left( u \right)\], hence we first find the function whose degree will be find
\[
p\left( u \right)q\left( u \right)r\left( u \right) = \left( {{u^2} + 3u + 4} \right)\left( {{u^2} + u - 12} \right)\left( {u - 2} \right) \\
= \left( {{u^4} + {u^3} - 12{u^2} + 3{u^3} + 3{u^2} - 36u + 4{u^2} + 4u - 48} \right)\left( {u - 2} \right) \\
= \left( {{u^4} + 4{u^3} - 5{u^2} - 32u - 48} \right)\left( {u - 2} \right){\text{ }}\left[ {\because {a^m} \times {a^n} = {a^{m + n}}} \right] \\
= \left( {{u^5} - 2{u^4} + 4{u^4} - 8{u^3} - 5{u^3} + 10{u^2} - 32{u^2} + 64u - 48u + 96} \right) \\
= \left( {{u^5} + 2{u^4} - 13{u^3} - 22{u^2} + 16u + 96} \right) \\
\]
So, the value of \[p\left( u \right)q\left( u \right)r\left( u \right) = \left( {{u^5} + 2{u^4} - 13{u^3} - 22{u^2} + 16u + 96} \right)\]
As we know the degree of a polynomial is the highest power of the nonzero coefficient monomial, so in the obtained polynomial we can see \[{u^5}\]has the highest degree with the coefficient 1 which is the order of the polynomial.
Hence, we can say the degree of \[p\left( u \right)q\left( u \right)r\left( u \right)\] is equal to 5.
Note: When two powers are multiplied together with the same base then their exponents adds up \[{a^m} \times {a^n} = {a^{m + n}}\] here base is \[a\] and the powers are m, n. Two powers with different bases cannot be added together.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE