Question

which method may be used to find the real root of the polynomial equation $p(x)=2x^{3}-x^{2}+x - 24 = 0$?
rational zero theorem & factor theorem.
factor by grouping.
none of the above.
all of the above.
integral zero theorem & factor theorem.
integral zero theorem & rational zero theorem & factor theorem.

Explanation:

The Rational Zero Theorem helps find possible rational roots of a polynomial. The Factor Theorem states that if \(P(c)=0\), then \((x - c)\) is a factor of \(P(x)\). For a polynomial like \(P(x)=2x^{3}-x^{2}+x - 24\), we can first use the Rational Zero Theorem to find potential rational roots and then the Factor Theorem to check if they are actual roots. Factor - by - grouping may not be straightforward for this polynomial as there is no obvious way to group the terms to factor. The Integral Zero Theorem is a special case of the Rational Zero Theorem for integer roots. So the Rational Zero Theorem and Factor Theorem are applicable methods.

Answer:

Rational Zero Theorem & Factor Theorem.