Question

Розкладіть на множники: 1) (5а² - 3b)² - 16a⁴; 3) (2a + 3b)²- (4a - 5b)²; 2) m⁸- (3c - 2m⁴)²; 4) (x - y + t)² - (x - y - t)²​

Answer 1

Ответ:

(5а²-3b)²-16a⁴=25a⁴+9b²-16a⁴=31a⁴+9b²

(2a+3b)²-(4a-5b)²=4a²+9b²-16a²-25b²=-12a²-41b²

m⁸-(3c-2m⁴)²=m⁸-9c²+4m⁸=5m⁸-9c²

(x-y+t)²-(x-y-t)²=x²+y²+t²-x²+y²+t²=2y²+2t²

Answer 2

1. (5a² - 3b)² - 16a⁴: To factorize this expression, let's first simplify the square term: (5a² - 3b)² = (5a² - 3b) × (5a² - 3b) Now, we can apply the difference of squares formula, which states that (a² - b²) = (a + b)(a - b): (5a² - 3b) × (5a² - 3b) - 16a⁴ = [(5a²)² - (3b)²] - 16a⁴ = (25a⁴ - 9b²) - 16a⁴ = 25a⁴ - 9b² - 16a⁴ = (25a⁴ - 16a⁴) - 9b² = 9a⁴ - 9b² The expression (5a² - 3b)² - 16a⁴ factors into 9a⁴ - 9b².

2. m⁸ - (3c - 2m⁴)²: Similarly, we can apply the difference of squares formula to the square term: (3c - 2m⁴)² = (3c - 2m⁴) × (3c - 2m⁴) = [(3c)² - (2m⁴)²] = (9c² - 4m⁸) Now, we can substitute this back into the original expression: m⁸ - (3c - 2m⁴)² = m⁸ - (9c² - 4m⁸) = m⁸ - 9c² + 4m⁸ = (m⁸ + 4m⁸) - 9c² = 5m⁸ - 9c² The expression m⁸ - (3c - 2m⁴)² factors into 5m⁸ - 9c².

3. (2a + 3b)² - (4a - 5b)²: Applying the difference of squares formula: (2a + 3b)² - (4a - 5b)² = [(2a + 3b) + (4a - 5b)][(2a + 3b) - (4a - 5b)] = [(2a + 3b + 4a - 5b)(2a + 3b - 4a + 5b)] = [(6a - 2b)(-2a + 8b)] The expression (2a + 3b)² - (4a - 5b)² factors into (6a - 2b)(-2a + 8b).

4. (x - y + t)² - (x - y - t)²: Applying the difference of squares formula: (x - y + t)² - (x - y - t)² = [(x - y + t) + (x - y - t)][(x - y + t) - (x - y - t)] = [(2x - 2y)(2t)] = 4(x - y)(t) The expression (x - y + t)² - (x - y - t)² factors into 4(x - y)(t).