Question

If $f(x) = {x^6} - 10{x^5} - 10{x^4} - 10{x^3} - 10{x^2} - 10x + 10$, then find the value of $f(11)$.
A. \[1\]
B. \[10\]
C. \[11\]
D. \[21\]

Answer 1

- Hint: While solving such a sum we substitute the value of \[x = 11\]in the equation i.e., wherever there is x in the equation we need to substitute it by \[11\] and solve it without any calculation error.
* BEDMAS stands for Brackets, Exponents, Division, Multiplication, Addition and Subtraction.
* Also, in a geometric progression, sum of first \[n\] terms is given by
${S_n} = a\left( {\dfrac{{{r^n} - 1}}{{r - 1}}} \right),r \ne 1$
Where, \[{S_n}\] means sum of first \[n\] terms, ‘\[a\]’ is the first term and ‘\[r\]’ is the common ratio of a geometric progression (G.P.).

Complete step-by-step solution -
Given, $f(x) = {x^6} - 10{x^5} - 10{x^4} - 10{x^3} - 10{x^2} - 10x + 10$
Since, we have to find the value of \[f(11)\], we will substitute the value of \[x = 11\] in the given equation.
Therefore, the given equation becomes $f(11) = {11^6} - 10 \times {11^5} - 10 \times {11^4} - 10 \times {11^3} - 10 \times {11^2} - 10 \times 11 + 10$
Now we will take 10 common from 2nd ,3rd ,4th ,5th and 6th term of RHS
$f(11) = {11^6} - 10({11^5} + {11^4} + {11^3} + {11^2} + 11) + 10$ …… (1)
Now if we observe the expression ${11^5} + {11^4} + {11^3} + {11^2} + 11$ carefully then we will realise that it is sum of terms (in descending order) of a geometric progression having the first term \[a = 11\] and common ratio \[r = 11\].
From the formula of sum of \[n\] terms of a G.P we have
${S_n} = a\left( {\dfrac{{{r^n} - 1}}{{r - 1}}} \right)$
Therefore, substituting the value of \[n = 5,a = 11,r = 11\]
${S_5} = 11\left( {\dfrac{{{{11}^5} - 1}}{{11 - 1}}} \right)$
 $ = 11\left( {\dfrac{{{{11}^5} - 1}}{{10}}} \right)$ { by using subtraction in the denominator}
Now we will multiply 11 inside the bracket
${S_5} = \left( {\dfrac{{{{11}^5} \times 11 - 1 \times 11}}{{10}}} \right)$
Since, we know when the base is same powers can be added, therefore we can write \[{11^5} \times 11 = {11^{5 + 1}}\]
${S_5} = \left( {\dfrac{{{{11}^{5 + 1}} - 11}}{{10}}} \right)$
 $ = \left( {\dfrac{{{{11}^6} - {{11}^1}}}{{10}}} \right)$
Therefore, we can write ${11^5} + {11^4} + {11^3} + {11^2} + 11 = \left( {\dfrac{{{{11}^6} - {{11}^1}}}{{10}}} \right)$
Now we will substitute the above obtained value of the expression in the equation (1)
$f(11) = {11^6} - 10 \times \left( {\dfrac{{{{11}^6} - 11}}{{10}}} \right) + 10$
$f(11) = {11^6} - \dfrac{{10 \times ({{11}^6} - 11)}}{{10}} + 10$
Cancel out the common factor \[10\] from the denominator and the numerator.
$f(11) = {11^6} - ({11^6} - 11) + 10$
Opening the brackets in the above equation.
$f(11) = {11^6} - {11^6} + 11 + 10$
Now we can cancel out the terms \[{11^6}, - {11^6}\] because they are of opposite sign
\[f(11) = 0 + 11 + 10 = 21\]
$f(11) = 21$
Hence, the value of $f(11) = 21$.

Thus, option D is correct.

Additional Information:
Zeros of Polynomial: It is a solution to the polynomial equation,\[P(x) = 0\]. It is that value of \[x\]that makes the polynomial equal to zero.
Factor of a polynomial: A non-zero polynomial \[g(x)\]is called a factor of any polynomial \[f(x)\]if and only if there exists some polynomial \[q(x)\]such that \[f(x) = g(x)q(x)\]

Note:
Students are likely to make mistakes while opening the brackets with negative signs along with them, always remember that multiplication with negative numbers changes the sign of the number from positive to negative and vice versa.
Alternative Method:
Another approach to solve the question would be.
Given, $f(x) = {x^6} - 10{x^5} - 10{x^4} - 10{x^3} - 10{x^2} - 10x + 10$
Since, we have to find the value of $f(11)$, we will substitute the value of \[x = 11\] in the given equation.
Therefore, the given equation becomes $f(11) = {11^6} - 10 \times {11^5} - 10 \times {11^4} - 10 \times {11^3} - 10 \times {11^2} - 10 \times 11 + 10$ …… (1)
Solve the equation 1 by applying BEDMAS rule
$f(11) = {11^6} - 10 \times {11^5} - 10 \times {11^4} - 10 \times {11^3} - 10 \times {11^2} - 10 \times 11 + 10$
We will first solve the exponent part.
$f(11) = 1771561 - 10 \times 161051 - 10 \times 14641 - 10 \times 1331 - 10 \times 121 - 10 \times 11 + 10$
Now we will solve the multiplication part.
$f(11) = 1771561 - 1610510 - 146410 - 13310 - 1210 - 110 + 10$
Now we will simplify the above equation.
$f(11) = 1771571 - 1610510 - 146410 - 13310 - 1210 - 110$
Now we will add all the numbers which are having negative sign as minus multiply by minus becomes plus and that is why we should add all those numbers which are having minus sign before them and after adding we will put bigger numbers sign before the sum of those numbers as the rule says we should put bigger number sign before the sum which is minus in the above equation
$f(11) = 1771571 - 1771550$
Now finally we will difference of the numbers in order to calculate $f(11)$
$f(11) = 21$
Hence, the value of $f(11) = 21$ if $f(x) = {x^6} - 10{x^5} - 10{x^4} - 10{x^3} - 10{x^2} - 10x + 10$ .
Thus, option D is correct.