Factor each of the following polynomials

Question

a. 𝑥^4 −16

b. 6𝑥^2 −7𝑥−5

Answer 1

x^4 -16 = (x^2+4)(x-2)(x+2)

that's as far as you can factor with real factors

but if you wanted to have imaginary factors, then

x^4-16 = (x+2i)(x-2i)(x-2)(x+2)

it's all based on the general formula a^2 -b^2 = (a+b)(a-b)

6x^2 -7x -5

=(2x+1)(3x-5) Look for integer factors of 6 and 5, trial & error to find ones that will give you -7. there's only a few possible combinations

Answer 2

a) x^4 - 16

this is the difference of two squares: x^2 and 4

to factor difference of two squares a^2 - b^2 is (a+b)(a-b)

so factorization would be (x^2 + 4)(x^2 - 4)

since x^2 - 4 is also the difference of two squares, further factors to:

(x^2 + 4)(x + 2)(x - 2)

b) 6x^2 - 7x - 5

find numbers that when multiplied is 6 * -5 = -30

and when added together are -7

those two numbers would be -10 and 3

break up 7x into -10x and 3x:

6x^2 + 3x - 10x - 5

factor by grouping:

3x(2x+1) - 5(2x+1)

distribute:

(3x - 5)(2x + 1)