Problem 61 - Elementary Algebra

Question

Factor completely. $$ x 3-12 x 2-x+12 $$

Step-by-Step Solution

Answer

Factored form: \((x - 12)(x + 1)(x - 1)\).

Step 1: Write the Polynomial

The polynomial given is: \[ x^3 - 12x^2 - x + 12 \]

Step 2: Group Terms

To factor this polynomial, we'll begin by grouping terms into pairs: \[ (x^3 - 12x^2) + (-x + 12) \]

Step 3: Factor Out the Greatest Common Factor (GCF) from Each Group

For the first group \((x^3 - 12x^2),\) factor out \(x^2:\)\[ x^2(x - 12) \]For the second group \((-x + 12),\) factor out \(-1:\)\[ -1(x - 12) \]

Step 4: Combine Common Factors

Now that both terms contain \((x - 12),\) you can factor it out:\[ (x - 12)(x^2 - 1) \]

Step 5: Factor Further

The term \((x^2 - 1)\) is a difference of squares. It can be factored further:\[ x^2 - 1 = (x + 1)(x - 1) \]Thus, the complete factorization is:\[ (x - 12)(x + 1)(x - 1) \]

Key Concepts

Polynomial Factorization

Polynomial factorization is all about breaking down a complex polynomial into simpler, multiplied forms. Think of it like dismantling a big puzzle into smaller, manageable pieces. The key goal is to express the polynomial as a product of its factors. This helps in simplifying the polynomial and solving equations easily.

When factoring a polynomial, it often helps to look for common terms or structures within each piece of the expression. In our example,

• the expression: \[ x^3 - 12x^2 - x + 12 \] can be seen as a mixture of terms that seem to have a shared structure, which we identify by grouping them efficiently.
• This requires a keen eye to pick out patterns where factors can be stripped out, much like spotting patterns within puzzles.

By grouping terms strategically,

• we simplify the expression
• making it easier to spot common elements and potential further factorization opportunities.

Difference of Squares

The difference of squares is a specific type of quadratic expression that takes the form \[ a^2 - b^2 \]. It’s one of the simpler forms of polynomials to factor, because it always breaks down into two linear factors: \[ (a + b)(a - b) \].

It's called the 'difference of squares' because each term is a perfect square, and they’re being subtracted. In our problem's final steps, after factoring out the greatest common factor, we encounter the expression:

• \[ x^2 - 1 \]

Here,

• \( x^2 \) is a perfect square (as it's \( x \cdot x \)),
• and 1 can be viewed as \( 1^2 \).
• Recognizing the pattern allowed us to express it as \[ (x + 1)(x - 1) \], thanks to this identity.

Utilizing the difference of squares is a handy shortcut for simplifying and solving polynomial expressions efficiently.

Greatest Common Factor

The greatest common factor (GCF) is all about finding the largest factor that two or more numbers share. In polynomial terms, it's the biggest expression that divides each term in a polynomial evenly.

In our step-by-step solution, the goal was to simplify the parts of the polynomial by pairing them, finding the GCF for each pair. Let's consider the steps:

• The first group is \( x^3 - 12x^2 \). Here, both terms share \( x^2 \) as a common factor.
• The second group is \(-x + 12\). Here, \(-1\) can be factored out.

By pulling out these common factors, the polynomial becomes simpler. Once factored, it reveals its underlying structure, and allows further factorization like recognizing other patterns such as a difference of squares. The GCF serves as a stepping stone in simplifying polynomials, making tricky expressions more approachable.