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Chapter 0: Problem 104

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

Short Answer

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

Step by step solution

01

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 \(-5(x-1)\).
02

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 \((x - 1)(12x^2 - 4x - 5)\).
03

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 \(4x(3x - 5) - 1(3x - 5)\).
04

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 \((x - 1)(3x - 5)(4x - 1)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Simplify by reducing the index of the radical. $$\sqrt[12]{x^{4} y^{8}}$$

Will help you prepare for the material covered in the next section.Exercises \(144-146\) will help you prepare for the material covered in the next section. Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. $$ \frac{x^{2}+6 x+5}{x^{2}-25} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) toproduce a true statement. $$534.7=5.347 \times 10^{3}$$

Evaluate each expression. $$\sqrt[3]{\sqrt{\sqrt{169}+\sqrt{9}}+\sqrt{\sqrt[3]{1000}+\sqrt[3]{216}}}$$

Why must \(a\) and \(b\) represent non negative numbers when we write \(\sqrt{a} \cdot \sqrt{b}=\sqrt{a b ?}\) Is it necessary to use this restriction in the case of \(\sqrt[3]{a} \cdot \sqrt[3]{b}=\sqrt[3]{a b} ?\) Explain.
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