Problem 31 - Elementary Algebra

Question

Factor. $$ 6 x 2-20 x-16 $$

Step-by-Step Solution

Answer

2(3x + 2)(x - 4)

Step 1: Identify Common Factors

Examine the coefficients in the expression \(6x^2 - 20x - 16\). All terms can be evenly divided by 2. Factor 2 out from all terms:\[6x^2 - 20x - 16 = 2(3x^2 - 10x - 8)\]

Step 2: Factor the Trinomial

We now need to factor the trinomial \(3x^2 - 10x - 8\). Start by finding two numbers that multiply to \(ac = -24\) (where \(a = 3\) and \(c = -8\)) and add up to \(b = -10\). These numbers are \(-12\) and \(2\).

Step 3: Split the Middle Term

Rewrite the middle term \(-10x\) using the numbers found: \(-12\) and \(2\).\[3x^2 - 12x + 2x - 8\]

Step 4: Group Terms

Group the expression into two pairs to make factoring by grouping possible:\[(3x^2 - 12x) + (2x - 8)\]

Step 5: Factor by Grouping

Factor out the greatest common factor from each group.For the first group \((3x^2 - 12x)\), factor out \(3x\):\[3x(x - 4)\]For the second group \((2x - 8)\), factor out \(2\):\[2(x - 4)\]Combine them:\[3x(x - 4) + 2(x - 4)\]

Step 6: Factor Out the Common Binomial

Notice that \((x - 4)\) is a common factor:\[(3x + 2)(x - 4)\]

Step 7: Combine with Common Factor

Multiply back the factor of 2 initially factored out:\[2(3x + 2)(x - 4)\]

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one operation (like addition or multiplication). These expressions can include terms such as constants, coefficients, and variables. Understanding algebraic expressions is crucial as they form the basis of algebra and other mathematical concepts. A simple example of an algebraic expression is \(3x + 2\). Here:

• The number 3 is called the coefficient, multiplying the variable \(x\).
• \(x\) is the variable, which can represent a range of values.
• The number 2 is a constant term.

When working with algebraic expressions, like in our example problem, you need to identify common factors, group terms strategically, and apply simplification strategies. This helps in finding equivalent forms or solutions, which is particularly useful when factoring or solving equations.

Quadratic Trinomials

A quadratic trinomial is a polynomial with three terms where the highest power of the variable is 2, represented as \(ax^2 + bx + c\). Factoring quadratic trinomials involves breaking them down into simpler binomials. In the problem \(3x^2 - 10x - 8\), it is a quadratic trinomial where:

• \(a = 3\)
• \(b = -10\)
• \(c = -8\)

To factor a trinomial, you look for two numbers that multiply to \(ac\) (the product of \(a\) and \(c\)) and add up to \(b\). This is the key to splitting the middle term and eventually factoring the expression. In this situation, the numbers -12 and 2 accomplish this, allowing us to rewrite the trinomial for further simplification.

Factor by Grouping

Factor by grouping is a method used to factor polynomials, especially useful for quartic or higher, and quadratic trinomials after splitting. This involves rearranging and grouping terms to make factoring easier.To employ factor by grouping, rewrite the expression such that the grouped parts contain a common factor. For the expression \(3x^2 - 12x + 2x - 8\), it becomes:

• Group as \((3x^2 - 12x) + (2x - 8)\)
• Factor out the GCF (Greatest Common Factor) in each group.

For \(3x^2 - 12x\), factor out \(3x\): \[3x(x - 4)\]For \(2x - 8\), factor out \(2\): \[2(x - 4)\]Now both groups have \((x - 4)\), which can be factored out, leaving: \[(3x + 2)(x - 4)\]. By combining strategic grouping and factoring out, the polynomial simplifies significantly, facilitating solutions or further operations.