Problem 58 - Beginning and Intermediate Algebra

Question

Factor completely. $$5 g^{2}+g-7$$

Step-by-Step Solution

Answer

The completely factored expression of \(5g^2 + g - 7\) is \((5g + 7)(g - 1)\).

Step 1: Identify the AC method components

In the given expression, \(5g^2 + g - 7\), we have \(a=5\), \(b=1\), and \(c=-7\). The AC method states that we should multiply \(a\) and \(c\), so we get: \(ac = 5 \times -7 = -35\).

Step 2: Find two numbers that multiply to \(ac\) and add to \(b\)

We need to find two numbers, \(m\) and \(n\), such that \(mn = ac = -35\) and \(m + n = b = 1\). After considering various factors of -35, we find that \(m = 7\) and \(n = -5\) satisfy these conditions, since \(7 \times -5 = -35\) and \(7 - 5 = 1\).

Step 3: Rewrite the expression with the new terms

Replace the middle term (\(g\)) with the two new terms found in the previous step (\(7g - 5g\)): \(5g^2 + 7g - 5g -7\).

Step 4: Factor by grouping

In this step, we'll factor the expression in pairs. Group the first two terms and the last two terms: \((5g^2 + 7g) - (5g + 7)\). Now, factor out the greatest common factor (GCF) from each group: \(g(5g + 7) - 1(5g + 7)\).

Step 5: Factor out the common binomial

We can now see that both terms have a common binomial factor \((5g + 7)\). Factor this out of each term: \((5g + 7)(g - 1)\). So, the completely factored expression for the given quadratic is \((5g + 7)(g - 1)\).

Key Concepts

AC method

The AC method is a fantastic technique for factoring quadratic equations, especially when the quadratic cannot be easily factored by inspection. The name 'AC method' comes from the idea of multiplying the coefficient of the quadratic term (\(a\)) with the constant term (\(c\)). With this approach, we transform the trinomial into a simpler form that can be factored by grouping.
For example, for the polynomial \(5g^2 + g - 7\), the values are \(a = 5\), \(b = 1\), and \(c = -7\).

• First, multiply the \(a\) term by the \(c\) term: \(ac = 5 \times -7 = -35\).
• Then, find two numbers that both multiply to \(-35\) and add to \(1\) (the \(b\) coefficient).
• Through trial and exploration, identify \(7\) and \(-5\) as these numbers, because \(7 \cdot -5 = -35\) and \(7 + -5 = 1\).

This clever technique helps in restructuring the polynomial for easier factorization moving forward.

Polynomial expressions

When working with polynomial expressions, you need to understand the structure and degree of the polynomial. A polynomial is a mathematical expression that consists of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents.
Quadratic polynomials, like \(5g^2 + g - 7\), are expressions of the form \(ax^2 + bx + c\). Here, the variable is \(g\), and it has a degree of 2, meaning the highest exponent on the variable is 2.

• The coefficients \(a\), \(b\), and \(c\) correspond to the terms in order of their power.
• If you are given a quadratic polynomial, your aim is often to factor it down into a product of simpler binomials, like turning \(5g^2 + g - 7\) into \((5g + 7)(g - 1)\).
• Factoring helps in simplifying expressions and solving equations easily as it reveals the roots or solutions of the polynomial equation when set to zero.

Understanding these basics make manipulating and solving polynomials more accessible and less intimidating.

Factor by grouping

Factor by grouping is an effective method for factoring polynomials, particularly when it comes to quadratics structured in a way where two pairs of terms can be grouped. This approach involves grouping terms in pairs and factoring out the greatest common factor from each pair.
To illustrate, use the expression \(5g^2 + 7g - 5g - 7\):

• First, split the expression into two groups: \((5g^2 + 7g) - (5g + 7)\).
• Next, factor out the greatest common factor from each of these groups: \(g(5g + 7)\) and \(-1(5g + 7)\).
• Once you factor each group, what's left is a common binomial factor, \((5g + 7)\).
• Finally, factor out this common factor to get \((5g + 7)(g - 1)\).

By using factor by grouping, the polynomial is reduced into simpler binomial terms which are easier to work with or solve. This method is particularly handy when polynomials do not seem to fit other straightforward factorization techniques.