Question

Fully factor the following polynomials. Explain your thought process with your solution. a. (x) = 2x³ − 25x² + 53x − 3

Answer 1

The fully factored form of the polynomial (f(x) = 2x³ - 25x² + 53x - 3) is (f(x) = (x - 1)(2x - 1)(x - 3)).

Explanation:

To factor (f(x) = 2x³ - 25x² + 53x - 3), we'll use the Rational Root Theorem and synthetic division to find the roots of the polynomial. The Rational Root Theorem states that any rational root of a polynomial equation is of the form
`\((p)/(q)\)`, where (p) is a factor of the constant term (-3 in this case) and (q) is a factor of the leading coefficient (2 in this case).

First, we check possible factors of -3: ±1, ±3. Synthetic division or direct substitution shows that (x = 1) is a root (yielding a remainder of 0). Therefore, (x - 1) is a factor.

Next, perform synthetic division by dividing (2x³ - 25x² + 53x - 3) by (x - 1) to find the quotient. The result is (2x² - 23x + 3).

To factor
`\(2x^2 - 23x + 3\)`, we can use quadratic factorization or the quadratic formula. The roots are (x = 3) and
`\(x = (1)/(2)\)`. Thus, (x - 3) and (2x - 1) are the remaining factors.

Therefore, combining all the factors, we get (f(x) = (x - 1)(2x - 1)(x - 3), representing the fully factored form of the given polynomial.