Answer:
- g(x) = (x -2)(x² +9) or g(x) = (x -2)(x -3i)(x +3i)
- real zero: x = 2
Explanation:
You want to factor g(x) = x³ -2x² +9x -18 and identify its real zeros.
Factor by grouping
Pairs of terms can be grouped together and factored. This will give a common factor that can be factored out. Then the remaining quadratic factor may or may not be factored.
g(x) = (x³ -2x²) +(9x -18)
g(x) = x²(x -2) +9(x -2)
g(x) = (x -2)(x² +9) . . . . . . . factored to real numbers
Difference of squares
The factoring of the difference of squares is ...
a² -b² = (a -b)(a +b)
The quadratic factor of g(x) can be considered to be the difference of squares, one of which is the square of an imaginary number.
x² +9 = x² -(-9) = x² -(3i)²
Factoring this to linear terms, we have ...
x² +9 = (x -3i)(x +3i)
so the complete factorization of g(x) to linear factors is ...
g(x) = (x -2)(x -3i)(x +3i)
Real zeros
The zeros of g(x) are the values of x that make the linear factors zero. The only real zero is the one that makes the real factor zero:
x -2 = 0 ⇒ x = 2 . . . . real zero of g(x)