Write 4/27 as the product of a prime raised to a power
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So focusing on x^4 + 5x^2 - 36, we will be completing the square. Firstly, what two terms have a product of -36x^4 and a sum of 5x^2? That would be 9x^2 and -4x^2. Replace 5x^2 with 9x^2 - 4x^2:
Next, factor x^4 + 9x^2 and -4x^2 - 36 separately. Make sure that they have the same quantity inside of the parentheses:
Now you can rewrite this as , however this is not completely factored. With (x^2 - 4), we are using the difference of squares, which is
. Applying that here, we have
. x^4 + 5x^2 - 36 is completely factored.
Next, focusing now on 2x^2 + 9x - 5, we will also be completing the square. What two terms have a product of -10x^2 and a sum of 9x? That would be 10x and -x. Replace 9x with 10x - x:
Next, factor 2x^2 + 10x and -x - 5 separately. Make sure that they have the same quantity on the inside:
Now you can rewrite the equation as . 2x^2 + 9x - 5 is completely factored.
