Final answer:
The question pertains to using the Rational Zero Theorem to find possible rational zeros for the polynomial P(x) = 3x⁴ - x³. The quadratic formula is used to solve quadratic equations, like 5x² - 7x - 8 = 0, by substituting the coefficients into the formula.
Step-by-step explanation:
Understanding the Rational Zero Theorem
The question involves the use of the Rational Zero Theorem for the polynomial P(x) = 3x⁴ - x³. When we refer to the Rational Zero Theorem, it specifically addresses possible rational zeros of a polynomial function. According to this theorem, if the polynomial has a rational zero p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then it will satisfy the equation P(x) = 0.
The polynomial given, 5x² - 7x - 8, can be solved using the quadratic formula. The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), is applied when we have a quadratic equation of the form ax² + bx + c = 0. It allows us to find the solutions for x by substituting the coefficients a, b, and c from the equation.
For the equation 5x² - 7x - 8 = 0, we can identify a = 5, b = -7, and c = -8 and substitute them into the quadratic formula to find the values of x. After performing the calculations, we obtain the possible zeros of the equation.
It is essential to always check the answer for reasonableness after performing algebraic manipulations. This includes evaluating whether the solutions make sense within the context of the problem and checking that the solutions satisfy the original equation.