To factor the quadratic polynomial \( 2x^2 - 5x - 3 = 0 \), we can use the method of factoring by grouping or the quadratic formula.
First, let's start by identifying the coefficients:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -3 \)
Next, we need to find two numbers that multiply to \( a \cdot c = 2 \cdot (-3) = -6 \) and add to \( b = -5 \). These two numbers are \( -6 \) and \( 1 \).
We can rewrite the middle term of the polynomial using these two numbers: \[ 2x^2 - 6x + x - 3 = 0 \]
Next, we can group the terms: \[ (2x^2 - 6x) + (x - 3) = 0 \]
Factor out the common factors from each group: \[ 2x(x - 3) + 1(x - 3) = 0 \]
Now, we can factor out the common binomial factor \( (x - 3) \): \[ (2x + 1)(x - 3) = 0 \]
Now, we can find the zeros by setting each factor to zero:
-
\( 2x + 1 = 0 \)
- Solving for \( x \): \[ 2x = -1 \implies x = -\frac{1}{2} \]
-
\( x - 3 = 0 \)
- Solving for \( x \): \[ x = 3 \]
Thus, the factored form of the polynomial is: \[ (2x + 1)(x - 3) = 0 \]
And the zeros (or roots) of the polynomial are: \[ x = -\frac{1}{2}, \quad x = 3 \]