# Math problem

Algebra

### Math problem

Hey can someone help me with some Algebra problem. i had 45 questions i solved 40 of them and i am left with 5 which i don't know how to solve.

https://ibb.co/t8n4cd6
Guest

### Re: Math problem

The first problem is
"Find the cubic polynomial f(x) such that f(2)= -1, f'(2)= -24, f''(2)= -26, and f'''(2)= -12".

It would have helped if you had told use what it was that was stopping you from solving tbe problem!

Do you know what a "cubic polynomial" is? Any cubic polynomial can be written in the form $$f(x)= ax^3+ bx^2+ cx+ d$$ for some number a, b, c, d. Finding f means finding those numbers. There are four so we need four equations or other conditions. And "f(2)= -1, f'(2)= -24, f''(2)= -26, and f'''(2)= -12" are precisely four conditions!

Do you know what f', f'', f''', and f'''' mean? Those are the first, second, third, and fourth derivatives:
f(2)= a(2^3)+ b(2^2)+ c(2)+ d= 8a+ 4b+ 2c+ d= -1.

f'(x)= 3ax^2+ 2bx+ c so
f'(2)= 3a(2^2+ 2b(2)+ c= 12a+ 4b+ c= -24.

f''(x)= 6ax+ 2b so
f''(2)= 12a+ 2b= -26

f'''(x)= 6a so
f'''(2)= 6a= -12.

Solve the four equations 6a= -12, 12a+ 2b= -26, 12a+ 4b+ c= -24, and 8a+ 4b+ 2c+ d= -1.

That is a "triangular system" so it is particularly easy to solve.
Solve 6a= -12 for a, put that value of a into 12a+ 2b= -26 and solve for b, put those values of a and b into 23a+ 4b+ c= -24 and solve for c, put those values of a, b, and c into 8a+ 4b+ 2c+ d= -1 and solve for d.
Guest

### Re: Math problem

Problem 2 doesn't actually say that A is a "linear transformation" but if it isn't this problem is impossible to solve. If it is then $$A(-2\vec{u}- 2\vec{v})= -2A(\vec{u})- 2A(\vec{v})$$ and you are told what A(u) and A(v) are!

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