Problem 21 - Algebra 2 and Trigonometry

Question

In \(9-26,\) write each expression as the product of two binomials. $$ 3 x^{2}-5 x-12 $$

Step-by-Step Solution

Answer

The expression \(3x^2 - 5x - 12\) factors to \((3x + 4)(x - 3)\).

Step 1: Identify the Quadratic Expression

The given quadratic expression is \(3x^2 - 5x - 12\). This expression is in the form \(ax^2 + bx + c\), where \(a = 3\), \(b = -5\), and \(c = -12\). Our task is to write this expression as a product of two binomials.

Step 2: Find Factors of ac

Multiply \(a\) and \(c\): \(ac = 3 \times -12 = -36\). We need to find two numbers that multiply to \(-36\) and add up to \(-5\) (the coefficient of the \(x\) term).

Step 3: Determine Correct Factor Pair

The two numbers that multiply to \(-36\) and add to \(-5\) are \(-9\) and \(4\) because \(-9 \times 4 = -36\) and \(-9 + 4 = -5\).

Step 4: Rewrite the Middle Term

Rewrite the expression \(3x^2 - 5x - 12\) by splitting the \(-5x\) into \(-9x + 4x\). This gives: \(3x^2 - 9x + 4x - 12\).

Step 5: Group Terms for Factoring

Group the terms in pairs: \((3x^2 - 9x) + (4x - 12)\). This helps in finding common factors.

Step 6: Factor Each Group

Factor each group separately. \(3x\) is a common factor in \(3x^2 - 9x\), and \(4\) is a common factor in \(4x - 12\). This gives: \(3x(x - 3) + 4(x - 3)\).

Step 7: Factor by Grouping

Note the common binomial factor \((x - 3)\). Thus, the expression becomes \((3x + 4)(x - 3)\).

Step 8: Verify Factored Form

Multiply \((3x + 4)(x - 3)\) to verify it returns to the original expression \(3x^2 - 5x - 12\). Expanding this, \(3x \times x + 3x \times (-3) + 4 \times x + 4 \times (-3) = 3x^2 - 9x + 4x - 12 = 3x^2 - 5x - 12\).

Step 9: State the Solution

Therefore, the expression \(3x^2 - 5x - 12\) can be written as the product of the binomials: \((3x + 4)(x - 3)\).

Key Concepts

Binomials

In mathematics, a binomial is an algebraic expression containing two different terms connected by a plus or minus sign. For example, the expression

• \(3x + 4\)
  
• \(x - 3\)

are both binomials. Binomials are the building blocks for factoring quadratic expressions because they allow us to break down complex equations into manageable parts.

The beauty of binomials lies in their simplicity and power to simplify and solve equations. When you express a quadratic equation as the product of two binomials, you have effectively factored it. Factoring makes it possible to solve equations and find roots or solutions easily.

Quadratic Expressions

A quadratic expression is a polynomial of degree two. It is in the standard form:

• \(ax^2 + bx + c\)

where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable.

Quadratic expressions are common in algebra and occur in various mathematical problems. In our exercise, the quadratic expression is \(3x^2 - 5x - 12\). Here, \(a = 3\), \(b = -5\), and \(c = -12\). The goal of factoring is to express this quadratic expression in terms of two simpler binomials.

By finding two numbers that multiply to the product of \(a\) and \(c\) (also known as \(ac\)) and add up to \(b\), we can rewrite the quadratic expression as a pair of binomials. This approach strategically simplifies the problem into a solution-ready format by decomposing the middle term, aiding in the factorization process.

Factoring by Grouping

Factoring by grouping is a method used to simplify and solve polynomial expressions. It involves grouping terms of the polynomial to find common factors, allowing the polynomial to be expressed as the product of simpler expressions.

In our exercise, after rewriting the quadratic expression \(3x^2 - 5x - 12\) as \(3x^2 - 9x + 4x - 12\), we need to group the terms to facilitate factoring:

• \((3x^2 - 9x)\)
• \((4x - 12)\)

Each group contains terms that can be factored further. For the first group, the common factor is \(3x\), and for the second group, it is \(4\). When factored, these groups become \(3x(x - 3)\) and \(4(x - 3)\) respectively.

By noticing the common binomial \((x - 3)\) in both groups, we can further factor the expression by grouping it into \((3x + 4)(x - 3)\), thereby simplifying the problem. This technique is crucial because it bridges the gap between recognizing patterns in numbers and finding a straightforward, solution-oriented approach to factoring complex expressions.