Answer
384.6k+ views
Hint: In this question, we have to find the value of a cubic polynomial at 6. Thus, we start solving this problem by letting a function which is equal to the subtraction of cubic polynomial at a variable and the variable. Then, we will find f(x), where x=0, 1, 2. After that, we know that a cubic equation can be expressed as $f(x)=a(x-1)(x-2)(x-3)$ . After that we will find P(4) and then P(6), to get the required result for the problem.
Complete step by step solution:
According to the question, we have to find the value of $P(6)$ of a cubic equation.
Let there be a function, such that it is equal to the subtraction of cubic polynomial at a variable and the variable, that is
$f(x)=P(x)-x$ --------- (1)
Thus, now we will find f(1), f(2), and f(3) from equation (1), we get
$\begin{align}
& f(1)=P(1)-1 \\
& f(1)=1-1 \\
& f(1)=0 \\
\end{align}$
Similarly,
$\begin{align}
& f(2)=P(2)-2 \\
& f(2)=2-2 \\
& f(2)=0 \\
\end{align}$ , and
$\begin{align}
& f(3)=P(3)-3 \\
& f(3)=3-3 \\
& f(3)=0 \\
\end{align}$
Thus, we get that $f(x)=0$ where x=1, 2, and 3.
Now, we will express the above equation in terms of a cubic polynomial as
$f(x)=a(x-1)(x-2)(x-3)$
$P(x)=a(x-1)(x-2)(x-3)+x$
Thus, $P(x)=f(x)+x$
$P(x)=a(x-1)(x-2)(x-3)+x$ -------- (2)
Now, we will put x = 4 and the value of P(4) from the question in equation (2), we get
$P(4)=a(4-1)(4-2)(4-3)+4$
On further simplification, we get
$5=a(3)(2)(1)+4$
On solving the brackets, we get
$5=6a+4$
Now, we will subtract 4 on both sides in the above equation, we get
$5-4=6a+4-4$
As we know, the same terms with opposite signs cancel out each other, thus we get
$1=6a$
So, we will divide 6 on both sides in the above equation, we get
$\dfrac{1}{6}=\dfrac{6}{6}a$
Thus, on further simplification, we get
$a=\dfrac{1}{6}$ ------------ (3)
Therefore, putting the value of equation (3) in equation (2), we get
$P(x)=\dfrac{1}{6}\left( (x-1)(x-2)(x-3) \right)+x$
Now, let x = 6 in the above equation, we get
$P(6)=\dfrac{1}{6}\left( (6-1)(6-2)(6-3) \right)+6$
On solving the brackets of the above equation, we get
$\begin{align}
& P(6)=\dfrac{1}{6}\left( (5)(4)(3) \right)+6 \\
& P(6)=\dfrac{1}{6}\left( 60 \right)+6 \\
\end{align}$
On further simplification, we get
\[\begin{align}
& P(6)=10+6 \\
& P(6)=16 \\
\end{align}\]
Therefore, the value of P(6) for the cubic polynomial is equal to 16.
Note: While solving this question, make all the calculations properly to avoid error and confusion. In this type of question, the accurate answer is based on the value of a, which we find out through P(4), so make all the steps properly.
Complete step by step solution:
According to the question, we have to find the value of $P(6)$ of a cubic equation.
Let there be a function, such that it is equal to the subtraction of cubic polynomial at a variable and the variable, that is
$f(x)=P(x)-x$ --------- (1)
Thus, now we will find f(1), f(2), and f(3) from equation (1), we get
$\begin{align}
& f(1)=P(1)-1 \\
& f(1)=1-1 \\
& f(1)=0 \\
\end{align}$
Similarly,
$\begin{align}
& f(2)=P(2)-2 \\
& f(2)=2-2 \\
& f(2)=0 \\
\end{align}$ , and
$\begin{align}
& f(3)=P(3)-3 \\
& f(3)=3-3 \\
& f(3)=0 \\
\end{align}$
Thus, we get that $f(x)=0$ where x=1, 2, and 3.
Now, we will express the above equation in terms of a cubic polynomial as
$f(x)=a(x-1)(x-2)(x-3)$
$P(x)=a(x-1)(x-2)(x-3)+x$
Thus, $P(x)=f(x)+x$
$P(x)=a(x-1)(x-2)(x-3)+x$ -------- (2)
Now, we will put x = 4 and the value of P(4) from the question in equation (2), we get
$P(4)=a(4-1)(4-2)(4-3)+4$
On further simplification, we get
$5=a(3)(2)(1)+4$
On solving the brackets, we get
$5=6a+4$
Now, we will subtract 4 on both sides in the above equation, we get
$5-4=6a+4-4$
As we know, the same terms with opposite signs cancel out each other, thus we get
$1=6a$
So, we will divide 6 on both sides in the above equation, we get
$\dfrac{1}{6}=\dfrac{6}{6}a$
Thus, on further simplification, we get
$a=\dfrac{1}{6}$ ------------ (3)
Therefore, putting the value of equation (3) in equation (2), we get
$P(x)=\dfrac{1}{6}\left( (x-1)(x-2)(x-3) \right)+x$
Now, let x = 6 in the above equation, we get
$P(6)=\dfrac{1}{6}\left( (6-1)(6-2)(6-3) \right)+6$
On solving the brackets of the above equation, we get
$\begin{align}
& P(6)=\dfrac{1}{6}\left( (5)(4)(3) \right)+6 \\
& P(6)=\dfrac{1}{6}\left( 60 \right)+6 \\
\end{align}$
On further simplification, we get
\[\begin{align}
& P(6)=10+6 \\
& P(6)=16 \\
\end{align}\]
Therefore, the value of P(6) for the cubic polynomial is equal to 16.
Note: While solving this question, make all the calculations properly to avoid error and confusion. In this type of question, the accurate answer is based on the value of a, which we find out through P(4), so make all the steps properly.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)