if f(x) = x^5+x^4-5x^3-2x^2+6x-2, g(x) = x^5-3x^3+2x^2-4x+2. How many distinct real numbers a satisfy f(a)=g(a)=0?

Answer 1

f(x) = g(x); x⁵ + x⁴ - 5x³ - 2x² + 6x - 2 = x⁵ - 3x³ + 2x² - 4x + 2; x⁴ - 2x³ - 4x² + 10x - 4 = 0.

List of possible rational zeros: ±1, ±2, ±4.

2 | 1 -2 -4 10 -4

2 0 -8 4

1 0 -4 2 |0 x = 2 is zero.

Quotient is q(x) = x³ - 4x + 2,This polinoial has 3 real zeros: between -3 and -2, between 0 and 1,

between 1 and 2 (Intermedium Value Theorem)

So, we have 4 real numbers a.

Answer 2

Looks like 3 based on just graphing the functions. However, they are all irrational, so it's not possible to find these analytically (by hand, synth division or factoring, etc.) so I don't really get the intention of the question. Unless it's just to get you to interpret it correctly that we are looking for shared x-intercepts of the 2 graphs.