- Extract the greatest common factor of 3 from the function \(f(x) = 36x^2 + 15x - 6\) to get \(3(12x^2 + 5x - 2)\).
- Factor the trinomial \(12x^2 + 5x - 2\) by finding two numbers that multiply to -24 and add to 5, which are 8 and -3.
- Split the middle term and factor by grouping to arrive at the expression \(3(4x - 1)(3x + 2)\).
- Compare this to the options provided and identify that it matches option \(\boxed{(C)}\).
Explanation
- Analyzing the Function
The function given is \(f(x) = 36x^2 + 15x - 6\). Our goal is to factor this quadratic function to identify its zeros and select the correct option from the given choices. We start by observing the coefficients: 36, 15, and -6. All of these are divisible by 3, which is the greatest common factor (GCF).
- Factoring the Quadratic Part
Factoring out the GCF of 3, we get:
\[f(x) = 3(12x^2 + 5x - 2)\]
Now, we need to factor the quadratic expression inside the parentheses, \(12x^2 + 5x - 2\). We look for two numbers whose product is \(12 \times (-2) = -24\) and whose sum is \(5\). These two numbers are 8 and -3, since \(8 \times (-3) = -24\) and \(8 + (-3) = 5\).
- Completing the Factorization
We can rewrite the middle term \(5x\) as \(8x - 3x\):
\[12x^2 + 5x - 2 = 12x^2 + 8x - 3x - 2\]
Grouping the terms, we have:
\[(12x^2 + 8x) - (3x + 2)\]
Factoring out common terms from each group:
\[4x(3x + 2) - 1(3x + 2)\]
Factoring out the common binomial \((3x + 2)\):
\[(4x - 1)(3x + 2)\]
Combining this back with the GCF of 3, the complete factored form is \(f(x) = 3(4x - 1)(3x + 2)\).
- Conclusion and Identifying the Correct Option
Comparing our result \(f(x) = 3(4x - 1)(3x + 2)\) with the given options:
A\(f(x) = (4x - 1)(3x + 2)\)
B\(f(x) = (12x - 2)(3x + 3)\)
C\(f(x) = 3(4x - 1)(3x + 2)\)
D\(f(x) = 3(12x - 2)(3x + 3)\)
Option (C) matches our derived factored form. Thus, the correct factored form is \(\boxed{(C)}\).
Examples
Factoring is essential in many real-world applications, such as physics and engineering, where we need to find the roots of a quadratic equation. For instance, if you are designing a path for a ball thrown into the air, the function representing its height over time is often a quadratic equation. Factoring that equation helps you determine exactly when the ball hits the ground by finding the zeros of the function, allowing for precise control and measurement in your design.