Answer:
2(n + 3)(n - 3)(n² + 4)
Step-by-step explanation:
Your polynomial is: P =2n⁴ - 10n² - 72
1. Remove the common factor
2n⁴ - 10n² - 72 = 2(n⁴ - 5n² - 36)
2. Factor the quadratic
P is a quadratic in n².
(a) Find two numbers that multiply to give -36 and add to give -5.
List the positive factors of 36: 1, 2, 3, 4, 6, 9, 18, 36
One of the numbers must be negative. Start with the numbers near the middle of the list.
By trial and error, you will find that -9 and +4 work:
-9 × 4 = 36 and -9 +4 = -5
(b) Rewrite -5n² as -9n² + 4n²
n⁴ - 9n² + 4n² - 36
(c) Factor the first two and the last two terms
n²(n² - 9) + 4(n² - 9)
(d) Separate the common factor
(n² - 9)(n² + 4)
So, P = 2(n² - 9)(n² + 4)
(e) Factor n² - 9
n² - 9 is the difference of two squares.
n² - 9 = (n + 3)(n - 3)
P = 2(n + 3)(n - 3)(n² + 4)
