Final answer:
The binomial factor of the expression (25x²+30x+9) is (5x+3). This is verified using the binomial theorem to expand (5x+3)², which results in the original expression.
Step-by-step explanation:
The student asked which binomial is a factor of the expression (25x²+30x+9). To find the factors of a quadratic expression like this, we can attempt to factor it directly. One of the methods to do this is by looking for two numbers that multiply to 25*9 (which is 225) and add up to 30. Those numbers are 15 and 15, which suggest that the binomial factor could be (5x+3), since (5x+3)*(5x+3) will give us the original expression.
We can verify this by applying the binomial theorem and expanding (5x+3)²:
- First term: (5x)² = 25x²
- Outer and Inner terms: 2*(5x)*(3) = 30x
- Last term: (3)² = 9
So, the expansion of (5x+3)² is indeed 25x²+30x+9, confirming that (5x+3) is a factor of the given expression.