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Given that [tex]\frac{2}{5}[/tex] is a zero of [tex]f(x) = 50x^3 + 25x^2 - 8x - 4[/tex], solve the polynomial equation:

[tex]50x^3 + 25x^2 - 8x - 4 = 0[/tex]

Find the solution set.

Answer :

Final answer:

To solve the given polynomial equation with a known zero of 2/5, use polynomial division, then find the remaining zeros through the quadratic formula or factoring. Factorization can also solve simpler equations like 2x^2 - 8 = 0, resulting in the solution set {-2, 2}.

Explanation:

To solve the polynomial equation 50x^3 + 25x^2 - 8x - 4 = 0 given that 2/5 is a zero, perform polynomial division by dividing the equation by x - (⅔). After simplifying, use the quadratic formula or factoring to find the other zeros. For example, if after the division we get a quadratic equation like ax^2 + bx + c, we can solve for x using the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a). Remember, when substituting numerical values into the equation, algebraic manipulations are preferred before making the substitution. This allows for easy adjustments if needed later on.

To solve another equation, such as 2x^2 - 8, we would set it equal to zero and factor it as 2(x^2 - 4) = 2(x - 2)(x + 2) = 0. The solution set in this case would be {-2, 2}.

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