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Question

1. Carlos factored the polynomial, as shown below. Explain the error and enter the correct solution.

(8x³ - 125) = (2x - 5)³

2. List all of the possible rational zeros of f(x) = 8x³ - 2x² + 4x + 4 by using the Rational Zero Theorem.

Answer 1

1. 8x³ - 125 = (2x)³ - 5³ = (2x - 5)(4x² + 10x + 25)
2. By Rational Zeros Theorem: factors of 4: ±1, ±2, ±4; factors of 8: ±1,±2,±4,±8

So, List of possible rational zeros: ±1, ±2,±4,±1/2, ±1/4, ±1/8.

No numbers from this list is zero of polynomial and because this polynomial has not rational zeros

And because we can't factor it this way

Answer 2

Carlos made the mistake of taking the cube root of each term of the polynomial(a common mistake). The cube root of 8x³ is 2x, likewise the cube root of 125 is 5. Look for the pattern, this a difference of perfect cubes:

(a³-b³) = (a-b)(a²+ab+b²) In our case a=2x, and b=5. Just plug into the equation.

8x³-125 = (2x-5)( (2x)²+(2x)(5)+(5)²) = (2x-5)(4x²+10x+25)

The second question, has to do with the rational root theorem. What it says that the only zeros have to be the constant term (last term) 4. divided by highest degree term (first term) 8. or factors of those. Factors of 4 are 4,2,1 divided by factors of 8 which is 8,4,2,1 so we have 4/8, 4/4,4/2,4/1 or 2/8, 2/4, 2/2, 2/1, or 1/8, 1/4, 1/2, 1/1 Now take out the duplicates we have possible rational roots 1/2, 1, 2, 4, 1/4, 1/8. Now each of these 6 possible roots we must consider the positive and negative ones always. +1/2 -1/2, +1, -1, +2,-2, +4, -4, +1/4 -1/4, +1/8, -1/8 plug them in and test to see if they make the equation zero to see if any are roots. It is possible that there are no rational roots, just complex roots.