show that k(x)= (5x^3) + (5/x^3) - x- (1/x)
^ sorry didn't finish: show that k(x)= k(1/x)
let me rewrite k(x)
k(x) = 5(x^3) + 5(1/x^3) - x/1 - 1/x
it should be intuitively obvious that
k(x) = k(1/x)
yea i realize that but the question says "show" that they equal each other. im thinking possibly by some sort of identity or something.
sub 1/x for x
k(1/x) = 5(1/x^3) + 5/(1/x^3) - 1/x - 1/(1/x)
= 5/x^3 + 5x^3 - 1/x - x
= the original k(x) , just the order of writing down the terms has changed.
ohh i see duh!! lol thanks =)
To show that k(x) = 5x^3 + 5/x^3 - x - 1/x, we can simplify the expression and demonstrate that the simplified form is equivalent to k(x).
1. Begin with the given expression: k(x) = 5x^3 + 5/x^3 - x - 1/x.
2. Simplify the expression by finding the least common denominator (LCD) for the fractions. In this case, the LCD is x^3.
- For the second term, 5/x^3, multiply both the numerator and denominator by x^3:
5/x^3 = 5(x^3)/x^3 = 5.
- Similarly, for the fourth term, 1/x, multiply both the numerator and denominator by x^3:
1/x = (x^3)/x^3 = x^2.
3. Substitute the simplified fractions back into the expression:
k(x) = 5x^3 + 5 - x - x^2.
4. Rearrange the terms to obtain the standard form of the polynomial:
k(x) = 5x^3 - x^2 + 5 - x.
Therefore, the expression 5x^3 + 5/x^3 - x - 1/x is equivalent to k(x) = 5x^3 - x^2 + 5 - x.