Question

Let \[p\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d\] . If \[p\left( 1 \right) = p\left( 2 \right) = p\left( 3 \right) = 0\] , then the value of \[p\left( 4 \right) + p\left( 0 \right)\] is
(A) \[10\]
(B) \[24\]
(C) \[25\]
(D) \[12\]

Answer 1

Hint: In order to find the value of \[p\left( 4 \right) + p\left( 0 \right)\] , first it is required to write the given polynomial expression in the form of the product of its factors and then substitute the values in the product of the factors of the polynomial to get the answer.

Complete step-by-step solution:
According to the question it is given that \[p\left( 1 \right) = p\left( 2 \right) = p\left( 3 \right) = 0\] , so it means that \[\left( {x - 1} \right),\left( {x - 2} \right),\left( {x - 3} \right)\] are the factors of the given polynomial \[p\left( x \right)\] .
So it is observed that the degree of the given polynomial is four. hence it has four roots.
So let the fourth root of the polynomial is \[\left( {x - m} \right)\] .
Hence, the given polynomial is written as:
 \[{x^4} + a{x^3} + b{x^2} + cx + d = \left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x - m} \right)\] ……………….(1)
After comparing the constant term in the above expressions, we get
 \[d = 6m\] .
So the value of m is written as:
 $ d = \dfrac{m}{6} $
Now in order to find the value of \[p\left( 4 \right) + p\left( 0 \right)\] , we first need to put the \[x = 4\] in the expression (1) and then solve it to get the value of \[p\left( 4 \right)\] .
After that, put \[x = 0\] again in the expression (1) and then solve it to get the value of \[p\left( 0 \right)\] .
Finally substitute the value of \[p\left( 0 \right)\& p\left( 4 \right)\] in the expression and hence solve the right hand side of the equation.
The value of \[p\left( 4 \right)\] is calculated as shown below:
 $ \Rightarrow $ \[p\left( 4 \right) = \left( {4 - 1} \right)\left( {4 - 2} \right)\left( {4 - 3} \right)\left( {4 - m} \right)\]
Now simplify the above expression as below:
 $ \Rightarrow $ \[p\left( 4 \right) = 3 \times 2 \times 1 \times \left( {4 - m} \right)\]
Now express the whole expression in the term of $ m $ .
 $ \Rightarrow $ \[p\left( 4 \right) = 24 - 6m\] .
The value of \[p\left( 0 \right)\] is calculated as shown below:
 $ \Rightarrow $ \[p\left( 0 \right) = \left( {0 - 1} \right)\left( {0 - 2} \right)\left( {0 - 3} \right)\left( {0 - m} \right)\]
Now simplify the above expression as shown below:
 $ \Rightarrow $ \[p\left( 0 \right) = - 1 \times \left( { - 2} \right) \times \left( { - 3} \right) \times \left( { - m} \right)\]
Now express the whole expression in the term of \[m\] .
 $ \Rightarrow $ \[p\left( 0 \right) = 6m\]
Now substitute the values of \[p\left( 0 \right)\& p\left( 4 \right)\] in the expression \[p\left( 4 \right) + p\left( 0 \right)\] as shown below:
 $ \Rightarrow $ \[p\left( 4 \right) + p\left( 0 \right) = 24 - 6m + 6m\]
Simplify the above expression.
 $ \Rightarrow $ \[p\left( 4 \right) + p\left( 0 \right) = 24\]
Therefore, the value of \[p\left( 4 \right) + p\left( 0 \right)\] is \[24\] .

Hence, the correct option is B.

Note: As we know that the factors of the polynomial are same as the degree of the polynomial if all the roots of the polynomial are real and distinct and the factors of the polynomial are less than the degree of the polynomial if some of the roots of the polynomial are imaginary.