SOLUTION: Assuming that 3u + v\neq0, simplify \dfrac{54u^2 v + 18uv^2}{9u} + 3v + \frac{5u^5 - 13u^2 v^2 + 7v^4}{3u + v}. Write the expression as a single fraction.

Algebra ->  Expressions-with-variables -> SOLUTION: Assuming that 3u + v\neq0, simplify \dfrac{54u^2 v + 18uv^2}{9u} + 3v + \frac{5u^5 - 13u^2 v^2 + 7v^4}{3u + v}. Write the expression as a single fraction.      Log On


   



Question 1209066: Assuming that 3u + v\neq0, simplify \dfrac{54u^2 v + 18uv^2}{9u} + 3v + \frac{5u^5 - 13u^2 v^2 + 7v^4}{3u + v}. Write the expression as a single fraction.
Answer by yurtman(42) About Me  (Show Source):
You can put this solution on YOUR website!
Let's break down the problem step by step:
**Step 1: Simplify the first two terms:**
$$\frac{54u^2 v + 18uv^2}{9u} + 3v = 6uv + 2v^2 + 3v$$
**Step 2: Combine like terms:**
$$6uv + 2v^2 + 3v = 2v^2 + 9uv$$
**Step 3: Combine the simplified expression with the third term:**
$$2v^2 + 9uv + \frac{5u^5 - 13u^2 v^2 + 7v^4}{3u + v}$$
**Step 4: Find a common denominator:**
To combine these fractions, we need a common denominator. The common denominator is $(3u + v)$. We need to multiply the first term by $\frac{3u + v}{3u + v}$:
$$\frac{(2v^2 + 9uv)(3u + v)}{3u + v} + \frac{5u^5 - 13u^2 v^2 + 7v^4}{3u + v}$$
**Step 5: Combine the numerators:**
$$\frac{6uv^2 + 27u^2v + 5u^5 - 13u^2 v^2 + 7v^4}{3u + v}$$
**Step 6: Combine like terms in the numerator:**
$$\frac{5u^5 + 14u^2v + 6uv^2 + 7v^4}{3u + v}$$
So, the simplified expression as a single fraction is:
$$\frac{5u^5 + 14u^2v + 6uv^2 + 7v^4}{3u + v}$$