Problem 104 - College Algebra

Question

Factor completely. $$ 12 x^{2}(x-1)-4 x(x-1)-5(x-1) $$

Step-by-Step Solution

Answer

The factorized form of the given polynomial is \((x - 1)(3x - 5)(4x - 1)\).

Step 1: Identify Common Factors

The first step in factorization is to identify common factors in all the terms of the given polynomial expression. In this case, the common factor is \((x-1)\), which exists in all of the terms \(12x^2(x-1)\), \(-4x(x-1)\) and \(-5(x-1)\).

Step 2: Factorize using the Common Factor

Now, factorize using the identified common factor. This is done by taking out the common factor from each term. \(12x^2(x-1) - 4x(x-1) - 5(x-1)\) becomes \((x - 1)(12x^2 - 4x - 5)\).

Step 3: Simplify the Polynomial

Now, simplify the polynomial \(12x^2 - 4x - 5\) within the result from Step 2. This polynomial can be factorized by looking for two numbers that multiply to (12*-5)=-60 and add up to -4. The numbers are -10 and 6. Therefore, \(12x^2 - 4x - 5\) becomes \(4x(3x - 5) - 1(3x - 5)\).

Step 4: Final Factorization

The polynomial has now been broken down to: \((x - 1)(4x(3x - 5) - 1(3x - 5))\). It can still be simplified by factoring out \((3x - 5)\) just like we did in Step 2. The final factorized form will then be \((x - 1)(3x - 5)(4x - 1)\).

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors, which can be simpler polynomials, numbers, or expressions. This process is analogous to breaking down a number into its prime components; for example, the number 28 can be expressed as the product of its prime factors: 2 x 2 x 7. Similarly, a polynomial like \(12x^2 - 4x - 5\) can be broken down into \(4x(3x - 5) - 1(3x - 5)\).

Factorization is an essential skill in algebra as it simplifies complex polynomials, making them easier to understand, manipulate, and solve. It's particularly useful in solving equations, simplifying fractions, and finding zeros of functions.

Common Factor

A common factor refers to an expression that is a factor of each term within a polynomial. Identifying a common factor simplifies the factorization process and is usually the first step when attempting to factor completely. In the example, \(12 x^{2}(x-1)-4 x(x-1)-5(x-1)\), the common factor is \(x-1\).

Once the common factor is identified, it's extracted from each term, which reduces the polynomial to a simpler form and often makes the remaining factors more apparent. This acts as a powerful tool for uncovering hidden patterns within polynomials and is a crucial step towards mastering algebraic manipulation.

Simplifying Polynomials

Simplifying polynomials involves reducing them to their simplest form by factoring out common factors and combining like terms. After extracting the common factor in a polynomial, like \(12x^2 - 4x - 5\), we look for expressions that can be multiplied together to give the original polynomial. In this case, \(12x^2 - 4x - 5\) is simplified into \(4x(3x - 5) - 1(3x - 5)\), highlighting another common factor, \(3x - 5\).

The goal of simplification is to express the polynomial in the most concise and manageable way possible. This can facilitate the solving of equations or inequalities involving polynomials, making them less intimidating and more solvable for students.