simplify the following expression and show your problem solving strategy
x cubed + x squared + x + 1 + 2(x cubed + x squared + x + 1) + 3(x cubed + x squared + x + 1) + ....100( x cubed + x squared + x + 1)
Wow... what a problem! Do you know how to distribute? Try that first with the 100(xcubed + x squared), and the other numbers next to paranpheses. Basically, distributing is just multiplying the number outside the paranpheses by each number inside the paranpheses. I think you then add all the x cubed and x squared ones up separetly. Hope I helped!
To simplify the given expression, let's first break it down into its constituent parts. We have:
x^3 + x^2 + x + 1
2(x^3 + x^2 + x + 1)
3(x^3 + x^2 + x + 1)
...
100(x^3 + x^2 + x + 1)
When we observe the repeating pattern, we notice that each term within the parentheses (x^3 + x^2 + x + 1) is the same in all the equations. So, we can factor it out as a common factor:
(x^3 + x^2 + x + 1)(1 + 2 + 3 + ... + 100)
Now, let's simplify the second part of the equation, which is the sum of consecutive numbers from 1 to 100. We can use the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term)
In this case, the first term (a1) is 1, the last term (an) is 100, and the number of terms (n) is 100. Substituting these values into the formula, we have:
Sum = (100/2)(1 + 100)
= 50(101)
= 5050
Now, let's substitute this simplified value back into the equation:
(x^3 + x^2 + x + 1)(5050)
Thus, the simplified expression is 5050 times the polynomial (x^3 + x^2 + x + 1):
5050(x^3 + x^2 + x + 1)