Answer:
K. 1
Explanation:
You want to know the number of x-intercepts of the graph of y = x³ -1.
Descartes rule of signs
When the expression is examined left to right, there is one sign change of the coefficients. The sign of x³ is positive, and the sign of -1 is negative. This means there is one positive real x-intercept (root).
When the signs of the odd degree terms are reversed, we have ...
y = -x³ -1
Both terms have a negative sign, so there are 0 sign changes, meaning there are 0 negative real x-intercepts (roots).
Factors
The factoring of the difference of two cubes is ...
a³ -b³ = (a -b)(a² +ab +b²)
For a=x and b=1, the factoring is ...
y = x³ -1 = (x -1)(x² +x +1)
The discriminant of the quadratic factor is ...
d = 1² -4(1)(1) = -3
Since it is negative, we know the roots of this factor are complex. The only x-intercept is the one contributed by the factor (x -1), which is zero for x=1.
There is one (1) x-intercept on the graph of y = x³ -1, choice K.
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Additional comment
The discriminant of the quadratic ax²+bx+c is d=b²-4ac. When it is negative, the quadratic has 2 complex roots.
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