Factor the Polynomial using factored form 3.2 (13)

Question

Factor the polynomial and use the factored form to find the zeros. (Enter your answers as a comma-separated list. Enter all answers using the appropriate multiplicities.)

P(x) = x⁴ − 8x² − 9

X=______?

Answer 1

I'm sure you have spent a lot of time and effort learning to factor quadratics. Can you see that x⁴ - 8x² - 9 can be thought of as a quadratic if you write it like this: (x²)² - 8x² - 9? In fact, you can make it even more clear if you make a substitution by letting w = x². You can then write the problem as w² - 8w - 9 which you can factor using your quadratic factoring skills into (w - 9)(w + 1). Once you get to this point, you can plug the x² back in (in place of the "w") to make it read (x² - 9)(x² + 1). Then you can see that the "x² - 9" is the difference of two squares and can be factored as (x + 3)(x - 3) resulting in the complete factored form of (x + 3)(x - 3)(x² + 1). To get the zeros, set it equal to zero. Then use the zero product property and set each factor equal to zero and solve.

Answer 2

Hi Tori,

P(x) = x⁴ − 8x² − 9 = 0 can be factored into (x² - 9)(x²+1). (x² - 9) can be further factored into (x+3)(x-3) (this is a difference of squares).

So, you have (x+3)(x-3)(x²+1). x= -3, 3 are the real zeroes. (x²+1) doesn't have any real solutions, only complex ones.