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Chapter 6: Problem 84

Factor by grouping $$4 x^{3}-x^{2}-12 x+3$$

Short Answer

Expert verified
The factorized form of the given polynomial is \((x^2 - 3) (4x - 1)\).

Step by step solution

01

Grouping

The polynomial \(4x^3-x^2-12x+3\) can be written in two groups as \((4x^3-x^2) + (-12x+3)\).
02

Factoring each group

Factorize each group separately. We get \(x^2(4x - 1) - 3(4x - 1)\) as the factored form.
03

Factoring the total expression

Now, our expression looks like \(A * B - C * B\). We can take \(B\) common; so our expression becomes \((x^2 - 3) * (4x - 1)\). So, the factored form of the given polynomial is \((x^2 - 3) * (4x - 1)\).

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Key Concepts

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

Grouping Method
The Grouping Method is a useful technique for factoring polynomials, especially when dealing with polynomials that have four terms. This method allows you to group terms to simplify and factor the polynomial. Here鈥檚 how it works:
  • **Identify Groups**: Divide the polynomial into two groups of terms. For example, in the polynomial \(4x^3 - x^2 - 12x + 3\), you can group terms as \((4x^3 - x^2) + (-12x + 3)\).
  • **Factor Each Group**: Within each group, look for common factors and simplify. The first group \((4x^3 - x^2)\) can be factored as \(x^2(4x - 1)\). The second group \((-12x + 3)\) can be rewritten as \(-3(4x - 1)\).
  • **Combine the Groups**: Once each group is factored, the expression can be reformed using a common factor from each group. This results in \((x^2 - 3)(4x - 1)\).
This method is particularly straightforward once you become familiar with spotting potential groupings and common factors, making it a reliable tool in your algebra toolkit.
Polynomial Expressions
Polynomial expressions comprise terms that involve variables raised to whole number powers. In the context of this exercise, the polynomial \(4x^3 - x^2 - 12x + 3\) is made up of four distinct terms:
  • \(4x^3\)
  • \(-x^2\)
  • \(-12x\)
  • \(3\)
Each of these terms has unique characteristics:
  • The coefficient, which is the number in front of the variable. For example, '4' in \(4x^3\).
  • The degree, determined by the power of the variable, such as 3 in \(4x^3\), indicating it's a cubic term.
  • Constant terms, like '3', that do not include variables and have a degree of zero.
Understanding these components helps breakdown and manipulate polynomials more easily. Recognizing the structure of polynomial expressions is vital when applying methods like factoring.
Algebraic Factoring
Algebraic factoring is the process of breaking down a polynomial into simpler, multiplicative components known as factors. It's a fundamental skill in algebra that simplifies expressions and solves polynomial equations.When factoring using the grouping method, as in \((x^2 - 3)(4x - 1)\), you essentially reverse multiply the binomials to confirm the initial polynomial:
  • **Expand**: Multiplying the factored form, \((x^2 - 3)(4x - 1)\), should yield the original polynomial.
  • **Verify**: Ensure that all terms from the expanded form match the original polynomial \(4x^3 - x^2 - 12x + 3\) to confirm your factoring is correct.
Algebraic factoring not only simplifies solving equations but also enhances understanding of polynomial identities. By mastering this skill, you gain a powerful tool for solving algebraic problems.

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Most popular questions from this chapter

Exercises \(150-152\) will help you prepare for the material covered in the next section. Evaluate \((3 x-1)(x+2)\) for \(x=\frac{1}{3}\)

Exercises \(137-139\) will help you prepare for the material covered in the next section. In each exercise, factor completely. $$3 x^{3}-75 x$$

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(y(y+8)=16(y-1)\)

Evaluate \(\frac{250 x}{100-x}\) for \(x=60\)

Solve each equation. \(3^{x^{2}-9 x+20}=1\)
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