Need help with several math problems. Thank you!

Question

f(x) = x^5-7x^4+19x^3-27x^2+20x-6

f(x) = x^3-5x^2+11x-15

(2x^3-x^2+x-3) / (x^2+3x-2)

Thank you!

Answer 1

f(x) = x^3 -5x^2 +11x -15 just try a few simple small integers, 0, 1, 2, 3. 3 works. x=3 is a zero. so x-3 is a factor. Use synthetic or long division to get the other factor

= (x-3)(x^2 -2x +5)

try completing the square, as the quadratic doesn't have integer factors.

x^2 -2x + 1 = -5 +1 = -4

(x-1)^2 =1(2)^2

x=1 + or - 2i

x = 1-2i or 1+2i

just one real zero

only one x intercept if you graph f(x)

(x^2-2x+5) factors into (x-1-2i)(x-1+2i) but normally you just stop with the integer real factors

f(x) = x^5 - 7x^4 + 19x^3 -27x^2 + 20x -6. Try 1,it works. one factor is x-1

(x-1)(x^4-6x^3+13x^2-14x+6). 1 works again

(x-1)^2(x^3-5x^2+8x^2-6)

the last one, the fraction might to be a typo somewhere, a sign or missing term?

usually you'd factor numerator and denominator, then cancel a common factor

but that doesn't seem to work. It's mostly trial and error finding factors

but if this is the real problem, about all you can do is

do long division

(2x^3-x^2+3)/(x^2+3x-2) = 2x-7 with a remainder of -17x-11