Find the zeroes of the following polynomials factorisation method and verify the relations between the zeroes and the coefficients of the polynomials.
`5t^(2)+12t+7`

Answer

To find the zeroes of the polynomial \(5t^2 + 12t + 7\) using the factorization method, we will follow these steps:

### Step 1: Identify the polynomial
The polynomial given is:
\[ 5t^2 + 12t + 7 \]

### Step 2: Multiply \(a\) and \(c\)
Here, \(a = 5\) and \(c = 7\). We need to calculate \(a \cdot c\):
\[ 5 \cdot 7 = 35 \]

### Step 3: Find two numbers that multiply to \(35\) and add to \(12\)
We need to find two numbers whose product is \(35\) and sum is \(12\). The numbers are \(7\) and \(5\) because:
\[ 7 \cdot 5 = 35 \quad \text{and} \quad 7 + 5 = 12 \]

### Step 4: Rewrite the middle term
We can rewrite the polynomial by splitting the middle term \(12t\) into \(7t + 5t\):
\[ 5t^2 + 7t + 5t + 7 \]

### Step 5: Factor by grouping
Now, we will group the terms:
\[ (5t^2 + 7t) + (5t + 7) \]
Now, factor out the common terms from each group:
\[ t(5t + 7) + 1(5t + 7) \]
This can be factored as:
\[ (5t + 7)(t + 1) \]

### Step 6: Set each factor to zero
To find the zeroes, we set each factor equal to zero:
1. \(5t + 7 = 0\)
\[ 5t = -7 \implies t = -\frac{7}{5} \]
2. \(t + 1 = 0\)
\[ t = -1 \]

### Step 7: List the zeroes
The zeroes of the polynomial are:
\[ t = -\frac{7}{5} \quad \text{and} \quad t = -1 \]

### Step 8: Verify the relationships between the zeroes and coefficients
The relationships are given by:
- Sum of the zeroes \( = -\frac{b}{a} \)
- Product of the zeroes \( = \frac{c}{a} \)

From our polynomial:
- \(a = 5\), \(b = 12\), \(c = 7\)

Calculating the sum of the zeroes:
\[ -\frac{b}{a} = -\frac{12}{5} \]
Calculating the sum of the zeroes we found:
\[ -\frac{7}{5} + (-1) = -\frac{7}{5} - \frac{5}{5} = -\frac{12}{5} \]
Thus, the sum of the zeroes matches.

Calculating the product of the zeroes:
\[ \frac{c}{a} = \frac{7}{5} \]
Calculating the product of the zeroes we found:
\[ -\frac{7}{5} \cdot (-1) = \frac{7}{5} \]
Thus, the product of the zeroes matches.

### Conclusion
The zeroes of the polynomial \(5t^2 + 12t + 7\) are \(t = -\frac{7}{5}\) and \(t = -1\). The relationships between the zeroes and the coefficients are verified.

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