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If \[p\left( u \right) = {u^2} + 3u + 4,q\left( u \right) = {u^2} + u - 12\]and\[r\left( u \right) = u - 2\], then find the degree of\[p\left( u \right)q\left( u \right)r\left( u \right)\].

Last updated date: 22nd Jun 2024
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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.