Problem 30 - Algebra 1: Concepts and Skills

Question

Factor the expression. \(8 x^{2}-3 x-8 x+3\)

Step-by-Step Solution

Answer

So, the factorized form of the expression \(8x^{2} -3x -8x + 3\) is \((x - 1) (8x - 3)\).

Step 1: Grouping Like Terms

Regroup the expression to combine like terms, which results in: \(8 x^{2} -8x -3x + 3\).

Step 2: Factor by Grouping

Factor by grouping means you group the terms that have common factors and remove the common factor. Here, apply the factorisation by grouping to get: \(8x(x - 1) -3 (x - 1)\). Note that \(8x\) and \(-3\) can be factored out of first and second binomials respectively.

Step 3: Factor Completely

Now, look at the entire expression. You'll notice that \(x - 1\) is a common factor in both terms which can be factored out. This results in: \((x - 1) (8x - 3)\).

Key Concepts

Factor by Grouping

The technique of factoring by grouping is used when an algebraic expression has no common factor across all terms but can be reorganized into groups that do. To factor an expression by grouping, you need to separate the terms into groups where each group has a common factor.

For example, in the expression \(8 x^{2}-3 x-8 x+3\), we can group the terms into two pairs: \(8 x^{2} - 8x\) and \( -3x + 3\). Once grouped, you can factor out the greatest common factor (GCF) from each pair. This process simplifies the expression to \(8x(x - 1) - 3(x - 1)\), where \(x - 1\) becomes evident as a common binomial factor that can be factored out in the next step.

Combining Like Terms

When working with algebraic expressions, it's essential to combine like terms to simplify the expression and see if factorization is possible. Like terms are terms that have the same variables raised to the same power. The only thing that can differ is the coefficient, the numerical part of the term.

In our example, \(8 x^{2}-3 x-8 x+3\), \(8 x^{2}\) stands alone with no like term, but \( -3 x\) and \( -8 x\) are like terms since they both have the variable \((x\)) raised to the power of one. By combining them, we get \(8 x^{2} - 8x - 3x + 3\), which is a necessary step before factoring by grouping.

Algebraic Factorization

The broader process of algebraic factorization involves rewriting an algebraic expression as a product of simpler factors. It is a vital tool in simplifying expressions and solving equations. The key is to identify patterns or common factors that can simplify the expression into a product of smaller terms.

After combining like terms, as in the expression \(8x(x - 1) - 3(x - 1)\), we recognize a common factor and use it to factor the entire expression. The final factored form, \(x - 1)(8x - 3)\), is a more straightforward representation that shows the roots of the equation if it were set to zero.

Common Factors

Identifying common factors is crucial for the factorization process. A common factor is a number or expression that divides a set of terms without any remainder. When you find common factors in two or more terms, it suggests that the expression can be factored further.

In the solution of our initial expression \(8x(x - 1) - 3(x - 1)\), \(x - 1\) is a common factor of both terms. By recognizing this, we can factor \(x - 1\) out from the expression, leading to a fully factored form of \(x - 1)(8x - 3)\). This skill is essential for simplifying algebraic expressions and solving equations.