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How to factor 12x^4-25x^3-8x^2 23x-6

Answer :

The factored form of the polynomial 12x^4 - 25x^3 - 8x^2 + 23x - 6 is (3x - 2)(2x + 1)(2x - 3)(x - 1).

To factor the given polynomial, we can use factoring techniques such as grouping and factoring by grouping.

Step 1: Group the terms in pairs and factor out the greatest common factor from each pair. We can factor out x^2 from the first two terms and factor out 1 from the last three terms:

x^2(12x^2 - 25x - 8) + 1(23x - 6)

Step 2: Factor the quadratic expression inside the parentheses. We are looking for two numbers that multiply to give -96 (product of the coefficients of x^2 and the constant term) and add up to -25 (coefficient of the linear term). The factored form is (3x - 8)(4x + 1).

Now, we have:

x^2(3x - 8)(4x + 1) + 1(23x - 6)

Step 3: Factor by grouping. We can factor out (3x - 2) from the first three terms and factor out 1 from the last two terms:

(3x - 2)(x^2(4x + 1) + 1(4x + 1))

Simplifying further, we get:

(3x - 2)(4x^3 + x^2 + 4x + 1)

Step 4: Factor the remaining expression. In this case, we can factor out (2x + 1) from the expression inside the parentheses:

(3x - 2)(2x + 1)(2x^2 - x + 1)

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