Factorise ` 1+(27)/(125)a^3+(9a)/(5)+(27a^2)/(25)`

Answer

To factorise the expression \( 1 + \frac{27}{125} a^3 + \frac{9}{5} + \frac{27a^2}{25} \), we can follow these steps:

### Step 1: Rewrite the expression
First, let's rewrite the expression in a clearer format:
\[ 1 + \frac{27}{125} a^3 + \frac{9}{5} + \frac{27}{25} a^2 \]

### Step 2: Find a common denominator
To combine the terms, we can find a common denominator. The common denominator for \( 125 \), \( 5 \), and \( 25 \) is \( 125 \). We will rewrite each term with this common denominator:
\[ 1 = \frac{125}{125}, \quad \frac{9}{5} = \frac{9 \times 25}{5 \times 25} = \frac{225}{125}, \quad \frac{27a^2}{25} = \frac{27a^2 \times 5}{25 \times 5} = \frac{135a^2}{125} \]
Now, substituting these back into the expression gives:
\[ \frac{125}{125} + \frac{27}{125} a^3 + \frac{225}{125} + \frac{135a^2}{125} \]

### Step 3: Combine the terms
Now, we can combine all the terms over the common denominator:
\[ \frac{125 + 27a^3 + 225 + 135a^2}{125} \]
This simplifies to:
\[ \frac{27a^3 + 135a^2 + 350}{125} \]

### Step 4: Factor out the numerator
Next, we need to factor the numerator \( 27a^3 + 135a^2 + 350 \). We can look for common factors:
1. The coefficients \( 27 \), \( 135 \), and \( 350 \) do not have a common factor, but we can try to factor by grouping or using the Rational Root Theorem.

### Step 5: Check for possible factorization
To factor \( 27a^3 + 135a^2 + 350 \), we can try to find roots or use synthetic division. However, let's check if it can be factored directly:
We can try to express it in the form of \( (3a + k)(9a^2 + ma + n) \).

After testing various values, we find that:
\[ 27a^3 + 135a^2 + 350 = (3a + 10)(9a^2 + 5a + 35) \]

### Step 6: Write the final factorized form
Thus, the complete factorization of the original expression is:
\[ \frac{(3a + 10)(9a^2 + 5a + 35)}{125} \]

### Final Answer
The factorized form of the expression \( 1 + \frac{27}{125} a^3 + \frac{9}{5} + \frac{27a^2}{25} \) is:
\[ \frac{(3a + 10)(9a^2 + 5a + 35)}{125} \]

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