2.jpeg "(a2+3a-2)2")
Expanding (a² + 3a - 2)²
This expression represents the square of a trinomial, (a² + 3a - 2). To expand it, we can apply the following methods:
1. Using the FOIL method
This method works by multiplying each term of the first binomial by each term of the second binomial. Since we're squaring the trinomial, we're effectively multiplying it by itself.
(a² + 3a - 2) * (a² + 3a - 2)
Step 1: Multiply the first terms of each binomial
- a² * a² = a⁴
Step 2: Multiply the outer terms of each binomial
- a² * 3a = 3a³
Step 3: Multiply the inner terms of each binomial
- 3a * a² = 3a³
Step 4: Multiply the last terms of each binomial
- -2 * -2 = 4
Step 5: Combine all the terms and simplify
- a⁴ + 3a³ + 3a³ + 9a² - 4a - 4a + 4
- a⁴ + 6a³ + 9a² - 8a + 4
2. Using the binomial theorem
The binomial theorem provides a general formula to expand expressions of the form (x + y)ⁿ. In this case, our "x" is the trinomial (a² + 3a - 2) and our "y" is 0.
(a² + 3a - 2)² = (a² + 3a - 2)⁰ * (a² + 3a - 2)²
We can then use the binomial theorem to expand the second term:
(a² + 3a - 2)² = ¹C₀ (a² + 3a - 2)² + ¹C₁ (a² + 3a - 2)¹ * 0 + ¹C₂ (a² + 3a - 2)⁰ * 0²
Simplifying this expression, we get:
(a² + 3a - 2)² = (a² + 3a - 2)²
We can now expand the square term using the FOIL method as described above, which results in the same final answer: a⁴ + 6a³ + 9a² - 8a + 4
Summary
The expansion of (a² + 3a - 2)² yields the polynomial a⁴ + 6a³ + 9a² - 8a + 4. We can achieve this result by using either the FOIL method or the binomial theorem. Both approaches lead to the same solution.