h(x)=\\frac{x^{2}-9x + 18}{x - 4}\nhow would you rewrite h(x) so it can be differentiated using the\npower…

Answer:

B. (x - 5-\frac{2}{x - 4})

Explanation:

Step1: Polynomial long division

Divide (x^{2}-9x + 18) by (x - 4). [ \begin{align*} x-4&\overline{\big)x^{2}-9x + 18}\ &x^{2}-4x\ &\overline{-5x+18}\ &-5x + 20\ &\overline{-2} \end{align*} ] So, (\frac{x^{2}-9x + 18}{x - 4}=x-5-\frac{2}{x - 4})

Step2: Power rule application

The power rule is (d/dx(x^{n})=nx^{n - 1}). For the function (y=x-5-2(x - 4)^{-1}), we can differentiate each term:

• The derivative of (x) is (1) (since (n = 1) in (x^{1}), (1\times x^{0}=1))
• The derivative of (-5) (a constant) is (0)
• For the term (-2(x - 4)^{-1}), using the power rule and chain rule ((d/dx(u^{n})=nu^{n - 1}\cdot u'), here (u=x - 4), (n=-1)), we have (-2\times(-1)(x - 4)^{-2}\times1=\frac{2}{(x - 4)^{2}})

If we use option A ((x^{2}-9x + 18)(x - 4)^{-1}), we would need to use the product rule ((uv)^\prime=u^\prime v+uv^\prime) (where (u=x^{2}-9x + 18), (v=(x - 4)^{-1})) instead of just the power rule. Option C (\frac{(x - 3)(x - 6)}{x - 4}) still has a quotient form that would require quotient rule or product rule (if rewritten as ((x - 3)(x - 6)(x - 4)^{-1})) rather than just power rule. Since we can rewrite (h(x)) as (x-5-\frac{2}{x - 4}) (sum of power - rule applicable terms: (x) ((n = 1)), constant ((n=0)) and ((x - 4)^{-1}) ((n=-1))), option D is incorrect.