Problem 17 - Elementary and Intermediate Algebra

Question

Complete each factorization. $$ 6 m^{2}+7 m-3=(-1)(2 m+1) $$

Step-by-Step Solution

Answer

The correct factorization is \((2m + 3)(3m - 1)\).

Step 1: Expand the Right-Hand Side

First, let's expand the expression on the right-hand side of the equation:\(-1(2m + 1) = -2m - 1\).

Step 2: Rearrange Equation

Rearrange the equation to compare coefficients by setting both sides equal. The original equation is:\(6m^2 + 7m - 3 = -2m - 1\).

Step 3: Correct the Factorization Form

Notice that the given factorization seems incorrect because the two sides aren't equal. Let's re-evaluate the original problem since the left and right sides don't match at all, meaning we likely need to find another factor to match the expression.

Step 4: Identify Potential Correct Factors

Since one factor does not exist on the right-hand side as expected from the problem, let's factor the left equation manually: \(6m^2 + 7m - 3\). Break the middle term to proceed factoring manually.

Step 5: Factor Manually

Consider factors of \(-18\) (the product of the leading coefficient \(6\) and \(-3\)) that add to \(7\). These factors are \(9\) and \(-2\). Rewrite the equation as:\(6m^2 + 9m - 2m - 3\).

Step 6: Group Terms

Group the terms to factor each pair separately:\((6m^2 + 9m) + (-2m - 3)\).

Step 7: Factor Each Group

Factor out the common factors in each group:\(3m(2m + 3) - 1(2m + 3)\).

Step 8: Factor Common Binomial

Notice that \((2m + 3)\) is common in both terms, so factor it out:\((2m + 3)(3m - 1)\).

Step 9: Completed Factorization

The fully factored form of the original polynomial is:\(6m^2 + 7m - 3 = (2m + 3)(3m - 1)\). This contradicts the given incorrect factor on the right side.

Key Concepts

Quadratic Equations

Quadratic equations are algebraic expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) represents an unknown variable. These equations are commonly used in mathematics to find values of \(x\) that satisfy the equation. In a quadratic equation:

• \(a\) is the coefficient of \(x^2\) and it should not be zero.
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

The solutions to a quadratic equation can be found using various methods such as factoring, completing the square, or using the quadratic formula.
For the problem, we have a quadratic expression \(6m^2 + 7m - 3\) where the goal was to factor it completely. Factoring is one powerful approach, particularly when the expression can be rewritten as the product of two binomials. The key is to rewrite the middle term and group strategically to uncover these binomials.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials. This technique is especially useful when direct factoring seems challenging. The process involves reorganizing the terms and finding common factors within groups, which allows it to be expressed as a product of simpler polynomials.For example, in the exercise given:- Start with the polynomial \(6m^2 + 7m - 3\).- Rewrite the middle term by picking two numbers that multiply to the product of the first and last coefficient (\(-18\)) and add to the middle coefficient \(7\). Here, these numbers are \(9\) and \(-2\).This gives us: \[6m^2 + 9m - 2m - 3\]Next, group the terms:\[(6m^2 + 9m) + (-2m - 3)\]Factor out common factors from each group:

• From \((6m^2 + 9m)\), factor out \(3m\) to get \(3m(2m + 3)\).
• From \((-2m - 3)\), factor out \(-1\) to get \(-1(2m + 3)\).

Finally, since \((2m + 3)\) is a common factor, factor it out:\[(2m + 3)(3m - 1)\]This yields the completely factored form of the polynomial. Factoring by grouping is beneficial in simplifying complex expressions and is especially handy for intermediate algebra students.

Polynomial Expressions

Polynomial expressions consist of variables raised to non-negative integer powers along with coefficients. The general form is \(a_nx^n + a_{n-1}x^{n-1} + \, ...\, + a_1x + a_0\). Each term combines a numerical coefficient and a variable part. Polynomials appear in various real-world contexts from computing areas, economics, to engineering.Properties of polynomials include:

• The degree of the polynomial, which is the highest power of the variable present.
• Coefficients that impact the shape and spread of the graph represented by the polynomial.
• Constant terms which represent horizontal shifts in the graph.

In the example of \(6m^2 + 7m - 3\), this is a quadratic polynomial where:

• The term \(6m^2\) represents the quadratic aspect, determining the parabola's direction and width.
• The term \(7m\) is linear, impacting the vertex's position.
• The \(-3\) is a constant term influencing the polynomial's y-intercept.

The goal in polynomial manipulation or factorization is to simplify expressions, solve equations, or graph functions effectively to understand their behaviors and relations.