Factoring these expressions

Question

Factor completely: 3x⁴+15x²

Determine the factors of: x²-13x+42

Factor completely: 25x²-36

Answer 1

1. 3X² (X² + 5)
2. ( X - 7 ) (X - 6 )
3. ( 5X + 6 ) ( 5X - 6)

Answer 2

For the first one, note that each term is divisible by both 3 and x². Thus, we pull out, or factor, a 3x² such that 3x⁴ + 15x² = 3x²(x² + 5).

For the second one, we need to find the two numbers that add to give -13 (the coefficient of the linear term) and multiply to give 42 (the constant), namely -6 and -7 (-6 + (-7) = -13 | (-6)*(-7) = 42). Then, we can write the factors as (x - 6) and (x - 7) such that x² - 13x + 42 = (x - 6)*(x - 7).

For the third one, you should notice that 25x² - 36 is the difference of two squares, or 25x² - 36 = (5x)² - (6)² for which there is a standard form for the factors. The difference of two squares a² - b² is factored as (a - b)*(a + b). Thus, in your case, 25x² - 36 = (5x - 6)*(5x + 6).