Question

Consider the polynomial expression, $ f(x)={{x}^{4}}+16{{x}^{3}}+72{{x}^{2}}+64x-129 $ , find the value of $ f(x-4) $ .

Answer 1

Hint: Replace $ x $ with $ (x-4) $ and use algebraic identities. Simplify it, and you will get the answer. Break the given polynomial into different terms and solve each term independently by replacing x by (x-4) which will make simplification easier.

Complete step-by-step answer:
Functions, in mathematics, can be an expression, a rule, or a law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
A technical definition of a function is: a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.
This means that if the object $ x $ is in the set of inputs (called the domain) then a function $ f $ will map the object $ x $ to exactly one object $ f(x) $ in the set of possible outputs (called the co domain).
This relationship is commonly symbolized as $ y=f(x) $ . In addition to $ f(x) $ , other abbreviated symbols such as $ g(x) $ and $ p(x) $ are often used to represent functions of the independent variable $ x $ , especially when the nature of the function is unknown or unspecified.
Let’s consider the example given in the question, $ f(x)={{x}^{2}} $ .
Here, if we take $ x=4 $ , we get the output as, $ f(4)={{4}^{2}}=16 $ .
So, an input of $ 4 $ gives $ 16 $ as the output.
Now, in the question, we have been given a function $ f(x)={{x}^{4}}+16{{x}^{3}}+72{{x}^{2}}+64x-129 $ , and we want to find $ f(x-4) $ .
We can thus see that $ x $ is simply replaced with $ (x-4) $ .
So, making the replacement, we get,
 $ f(x-4)={{(x-4)}^{4}}+16{{(x-4)}^{3}}+72{{(x-4)}^{2}}+64(x-4)-129 $ ………. (1).
Now let’s take the separate terms and simplify them one by one.
First, let us take $ {{(x-4)}^{4}} $ and write it in the form $ {{({{(x-4)}^{2}})}^{2}} $ .
We know the identity $ {{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}} $ .
Applying the above identity we get,
 $ {{({{(x-4)}^{2}})}^{2}}={{\left( {{x}^{2}}-8x+16 \right)}^{2}} $
So now solving $ {{\left( {{x}^{2}}-\left( 8x-16 \right) \right)}^{2}} $ , using the formula $ {{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}} $ , we get,
 $ \begin{align}
  & {{\left( {{x}^{2}}-(8x-16) \right)}^{2}}=\left( {{x}^{4}}-2({{x}^{2}})(8x-16)+{{(8x-16)}^{2}} \right) \\
 & \Rightarrow \left( {{x}^{4}}-2({{x}^{2}})(8x-16)+{{(8x-16)}^{2}} \right)=\left( {{x}^{4}}-16{{x}^{3}}+32{{x}^{2}}+64{{x}^{2}}-256x+256 \right) \\
\end{align} $
Simplifying further, we get,
 $ \left( {{x}^{4}}-16{{x}^{3}}+96{{x}^{2}}-256x+256 \right) $
So, we have,
 $ {{(x-4)}^{4}}=\left( {{x}^{4}}-16{{x}^{3}}+96{{x}^{2}}-256x+256 \right) $ ……………(2)
Now let’s take the next term $ {{(x-4)}^{3}} $ .
We know the identity, $ {{(a-b)}^{3}}={{a}^{3}}-{{b}^{3}}-3{{a}^{2}}b+3a{{b}^{2}} $ .
Using the above identity for $ {{(x-4)}^{3}} $ we get,
 $ {{(x-4)}^{3}}={{x}^{3}}-{{4}^{3}}-3{{x}^{2}}(4)+3x{{(4)}^{2}} $
So simplifying we get,
 $ {{(x-4)}^{3}}={{x}^{3}}-{{4}^{3}}-12{{x}^{2}}+48x $ .
Rearranging we get,
 $ {{(x-4)}^{3}}={{x}^{3}}-12{{x}^{2}}+48x-64 $ ……………. (3)
Now taking $ {{(x-4)}^{2}} $ , we get,
 $ {{(x-4)}^{2}}={{x}^{2}}-8x+16 $ …………………. (4)
Now substituting (4), (3), (2) in (1) we get, $ f(x-4)=\left( {{x}^{4}}-16{{x}^{3}}+96{{x}^{2}}-256x+256 \right)+16({{x}^{3}}-12{{x}^{2}}+48x-64)+72({{x}^{2}}-8x+16)+64(x-4)-129 $
So now simplifying the above equation, we get,
 $ f(x-4)=\left( {{x}^{4}}-16{{x}^{3}}+96{{x}^{2}}-256x+256 \right)+16{{x}^{3}}-192{{x}^{2}}+768x-1024+72{{x}^{2}}-576x+1152+64x-256-129 $
Rearranging the above we get,
 $ f(x-4)={{x}^{4}}-16{{x}^{3}}+16{{x}^{3}}+96{{x}^{2}}-192{{x}^{2}}+72{{x}^{2}}-256x-576x+768x+64x-1024+1152-256+256-129 $
Simplifying further can grouping like terms, we get,
 $ f(x-4)={{x}^{4}}-24{{x}^{2}}-1 $ .
So, the final answer we get is $ f(x-4)={{x}^{4}}-24{{x}^{2}}-1 $ .

Note: You should know the identities such as $ {{(a-b)}^{2}},{{(a-b)}^{3}} $ etc. You must also know how to solve the expression $ {{(a-b)}^{4}} $ . Be thorough with the identities. While simplifying don’t get confused. Sometimes, what happens is that the numbers or powers go missing. So take care of it.
Simplifying these $ f(x-4)=\left( {{x}^{4}}-16{{x}^{3}}+96{{x}^{2}}-256x+256 \right)+16{{x}^{3}}-192{{x}^{2}}+768x-1024+72{{x}^{2}}-576x+1152+64x-256-129 $ is not a big deal but mistakes occur when you try to simplify it.