`2z^(2)-13z +15 `=________

Options

A. `(z-5)(2z-3)`
B. `(x+5)(2z-3)`
C. `(z+5)(2z+3)`
D. `(z-5)(2z+3)`

Answer

To factorize the expression \(2z^2 - 13z + 15\), we will follow these steps:

### Step 1: Identify the coefficients
The expression is in the form \(az^2 + bz + c\), where:
- \(a = 2\)
- \(b = -13\)
- \(c = 15\)

### Step 2: Multiply \(a\) and \(c\)
We need to multiply \(a\) and \(c\):
\[ a \cdot c = 2 \cdot 15 = 30 \]

### Step 3: Find two numbers that multiply to \(30\) and add to \(-13\)
We need to find two numbers that multiply to \(30\) and add to \(-13\). The numbers are \(-10\) and \(-3\) because:
\[ -10 \cdot -3 = 30 \quad \text{and} \quad -10 + (-3) = -13 \]

### Step 4: Rewrite the middle term
Now we can rewrite the expression \(2z^2 - 13z + 15\) using \(-10z\) and \(-3z\):
\[ 2z^2 - 10z - 3z + 15 \]

### Step 5: Factor by grouping
Next, we will group the terms:
\[ (2z^2 - 10z) + (-3z + 15) \]
Now, factor out the common factors in each group:
\[ 2z(z - 5) - 3(z - 5) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \((z - 5)\):
\[ (2z - 3)(z - 5) \]

### Final Answer
Thus, the factorization of the expression \(2z^2 - 13z + 15\) is:
\[ (2z - 3)(z - 5) \]

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