Factoring the Trinomial: 5x² + 8x + 4
To factor the trinomial 5x² + 8x + 4, we need to find two binomials that, when multiplied together, yield the original trinomial. Let’s break down the process.
Step 1: Identify the Coefficients
The trinomial consists of three parts: the coefficient of x² (which is 5), the coefficient of x (which is 8), and the constant term (which is 4). This means we can express our trinomial in the form:
5x² + 8x + 4 = (ax + b)(cx + d)
where a, b, c, and d are values we need to determine.
Step 2: Trial and Error with Factors of the Constant
To find these values, we can start by analyzing the possible pairs of factors of the constant term (4). The pairs of factors of 4 are:
- (1, 4)
- (2, 2)
Step 3: Find Combinations of Coefficients
Next, we need to factor out the leading coefficient, which is 5. The possible factors of 5 are 1 and 5. Our pairs of factors for the trinomial can be:
- Factor pairs using (1, 4): (1, 1) and (5, 4)
- Factor pairs using (2, 2): (2, 1) and (5, 2)
Step 4: Applying the FOIL Method
Now, we combine these pairs:
- For (1, 4):
(5x + 4)(x + 1) - For (2, 2):
(5x + 2)(x + 2)
Both combinations can be factored further, and we find:
Step 5: Verification
Using the FOIL (First, Outside, Inside, Last) method, we can verify if these combinations yield the original trinomial:
- (5x + 4)(x + 1) = 5x² + 9x + 4
- (5x + 2)(x + 2) = 5x² + 8x + 4
Conclusion
From the verification, we can conclude that the factors which can be multiplied together to create the trinomial 5x² + 8x + 4 are:
(5x + 2)(x + 2).