What's the easiest way to factor this completely?
2x^4-10x^2-72
A step-by-step solution would really help me a lot. Thanks :)
2(x^4-5x^2-36)
2(x^2+4)(x^2-9)
2(x^2+4)(x-3)(x+3)
To factor the polynomial 2x^4 - 10x^2 - 72 completely, we can follow these steps:
Step 1: Look for common factors
First, check if there are any common factors in all the terms. In this case, the terms do not have any common factors other than 2, so we can factor out 2 as a common factor:
2(x^4 - 5x^2 - 36)
Step 2: Factor the quadratic expression inside the parentheses
Now, we need to factor the quadratic expression x^4 - 5x^2 - 36 inside the parentheses. To do this, we can treat it as a quadratic in terms of x^2. Let y = x^2, so our equation becomes:
y^2 - 5y - 36
To factor this quadratic expression, we need to find two numbers that add to -5 and multiply to -36. The numbers that satisfy these conditions are -9 and 4. Therefore, we can rewrite the equation as:
(y - 9)(y + 4)
Step 3: Substitute back for y
Now, substitute back y = x^2 into the factored quadratic expression:
(x^2 - 9)(x^2 + 4)
Step 4: Factor the difference of squares
To factor x^2 - 9 and x^2 + 4, we can use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). Applying this formula, we get:
(x - 3)(x + 3)(x^2 + 4)
Therefore, the completely factored form of 2x^4 - 10x^2 - 72 is:
2(x - 3)(x + 3)(x^2 + 4)
And that's the final answer.