f(x)=4x³-8x²-x+2
I'm having trouble factorising this one completely. I know that x-2 is a factor.
Please help!
4x2(x - 2) - (x - 2)
(4x2 - 1)(x - 2}
difference of two squares!
(2x - 1)(2x + 1)(x - 2)
To factorize the polynomial completely, you can use the method of synthetic division to divide the polynomial by the known factor. Here's how you can factorize the polynomial f(x) = 4x³ - 8x² - x + 2 completely:
Step 1: Given that x - 2 is a factor, you can use synthetic division to divide the polynomial by x - 2:
2 │ 4 -8 -1 2
------------------
8 -16 -34 -60
The numbers at the bottom of the synthetic division will give you the coefficients of the polynomial generated after the division.
Step 2: The result of the synthetic division is: 4x² - 16x - 34 with a remainder of -60.
Step 3: Now you have a quadratic polynomial that can be factored. To find the factors of the quadratic, you can use the quadratic formula or try to factor it further. In this case, since the quadratic cannot be factored further easily, we will use the quadratic formula:
The quadratic formula states that for an equation ax²+bx+c=0, the solutions are given by:
x = (-b ± √(b²-4ac)) / 2a
For the quadratic equation 4x² - 16x - 34 = 0, the values of a, b, and c are 4, -16, and -34, respectively. Substituting these values into the formula, you can calculate the solutions for x:
x = (-(-16) ± √((-16)² - 4(4)(-34))) / (2(4))
x = (16 ± √(256 + 544)) / 8
x = (16 ± √800) / 8
x = (16 ± 28.28) / 8
So the solutions are x = (16 + 28.28) / 8 = 5.29 or x = (16 - 28.28) / 8 = -1.09.
Hence, the factored form of the given polynomial f(x) = 4x³ - 8x² - x + 2 is f(x) = (x - 2)(2x - 5.29)(x + 1.09).
I hope this helps! Let me know if you have any further questions.