Problem 71 - Introductory Algebra for College Students

Question

Use the negative of the greatest common factor to factor completely. $$-x^{2}-3 x+40$$

Step-by-Step Solution

Answer

The completely factored form of the given equation is \(-1*(x - 5)(x + 8)\).

Step 1: Identify the quadratic equation

The quadratic equation to be factored is \(-x^{2}-3x+40\). This equation can be written as \(ax^{2} + bx + c\) where \(a=-1\), \(b=-3\), and \(c=40\).

Step 2: Check for greatest common factor

The greatest common factor (GCF) for the coefficients in the equation is 1. The negative of this is -1.

Step 3: Apply the negative of the GCF

Applying the negative of the GCF to the equation, we get \(-(-x^{2}-3x+40) = x^2 + 3x - 40\).

Step 4: Factor the quadratic equation

We now need to factor the equation \(x^{2} + 3x - 40\). The factors of 40 which add up to 3 are 8 and -5. Hence this equation can be factored as \((x - 5)(x + 8)\).

Step 5: Write the complete factored form

Therefore, the completely factored form of the given equation is \(-1*(x - 5)(x + 8)\).

Key Concepts

Greatest Common Factor

Understanding the greatest common factor (GCF) is essential when trying to factor polynomial expressions, particularly quadratic equations. The GCF is the highest number that divides exactly into two or more numbers. When we're looking at an algebraic expression, the GCF is the highest expression that is a factor of each term.

For instance, consider the equation \( -x^2 - 3x + 40 \). At first glance, the GCF of the numerical coefficients (1, 3, and 40) appears to be 1, but it's crucial to identify any common variable factors as well. In this case, since there are no common variable factors and the coefficients share no greater common divisor, the numeric GCF remains 1. However, the presence of a negative sign allows us to use \( -1 \) as the GCF, which can then be factored out to simplify the expression and prepare it for further factoring steps.

• Identify numerical and variable factors in all terms of the expression.
• Determine the GCF, considering both types of factors.
• Factor out the GCF from the expression before proceeding to the next factoring steps.

Using the GCF efficiently can significantly simplify the process of factoring complex polynomials, leading to a more streamlined solution.

Quadratic Formula

When factoring quadratic equations proves challenging or when quadratics do not factor easily, the quadratic formula comes into play as a powerful tool. The quadratic formula states that for any quadratic equation of the form \( ax^2 + bx + c = 0 \), the values of x can be found using: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]

This formula is derived from the process of completing the square and provides the roots of the equation directly. The discriminant, \( b^2 - 4ac \), determines the nature of the roots. If it is positive, there are two distinct real roots; if zero, there is one real root; and if negative, there are two complex roots.

• Substitute the coefficients \(a\), \(b\), and \(c\) into the formula.
• Calculate the discriminant first to gauge the number and type of roots.
• Compute the roots using the ± sign to develop both possible solutions.

By mastering the quadratic formula, students can confidently solve any quadratic equation, regardless of whether it is factorable or not.

Polynomial Factoring

Polynomial factoring is a critical step in solving quadratic equations and higher degree polynomials. Factoring involves rewriting the polynomial as a product of its simpler, irreducible factors. For a quadratic equation such as \( ax^2 + bx + c \), the goal is often to decompose it into a product of two binomials: \[ (dx + e)(fx + g) = ax^2 + bx + c \]

The process begins by identifying two numbers that multiply to give \(ac\) and add up to \(b\), which are then used to split the middle term and factor by grouping. In the context of the sample equation \( -x^2 - 3x + 40 \), we reverse the sign using the GCF \( -1 \) and then find factors of \( -40 \) that add up to 3, giving us 8 and -5. The equation can then be factored into \( (x - 5)(x + 8) \).

• Seek a pair of numbers with the needed multiplicative and additive properties relative to \(ac\) and \(b\).
• Use those numbers to split the middle term if applicable.
• Factor by grouping or by applying special factoring rules such as difference of squares or perfect square trinomials.

With practice, students can develop an intuition for the factoring process, making it easier to tackle more complex equations.