Problem 38 - Elementary and Intermediate Algebra

Question

Factor. See Example 3 or Example 6. $$ 5 y^{2}-18 y-8 $$

Step-by-Step Solution

Answer

The factored form is \((y - 4)(5y + 2)\).

Step 1: Identify the Quadratic Coefficient, Linear Coefficient, and Constant

In the quadratic expression \(5y^{2} - 18y - 8\), list out the coefficients: \(a = 5\), \(b = -18\), and \(c = -8\). These coefficients will be used to find the factors of the expression.

Step 2: Calculate the Product \(ac\)

Multiply \(a\) and \(c\) to find \(ac\). Here, \(a = 5\) and \(c = -8\), so \(ac = 5 \times -8 = -40\).

Step 3: Find Two Numbers that Multiply to \(ac\) and Add to \(b\)

We need to find two numbers that multiply to \(-40\) (the product) and add up to \(-18\) (the coefficient of \(y\)). The numbers that satisfy this are \(-20\) and \(+2\): \(-20 \times 2 = -40\) and \(-20 + 2 = -18\).

Step 4: Write the Middle Term Using the Two Numbers Found

Rewrite the expression \(5y^{2} - 18y - 8\) by splitting the middle term, \(-18y\), into \(-20y + 2y\). The expression becomes \(5y^{2} - 20y + 2y - 8\).

Step 5: Factor by Grouping

Take the expression \(5y^{2} - 20y + 2y - 8\). Group the first two terms and the last two terms: \((5y^{2} - 20y) + (2y - 8)\). Factor out the greatest common factor (GCF) from each group. For the first group, the GCF is \(5y\): \(5y(y - 4)\). For the second group, the GCF is \(2\): \(2(y - 4)\). The expression becomes \(5y(y - 4) + 2(y - 4)\).

Step 6: Factor Out the Common Binomial

Now that each term contains the common factor \((y - 4)\), factor it out to get \((y - 4)(5y + 2)\). This is the factored form of the original expression.

Key Concepts

Quadratic Coefficient

In the world of quadratic expressions, the term "quadratic coefficient" is used to refer to the coefficient of the squared term. In our example of factoring the expression \(5y^{2} - 18y - 8\), this is the term associated with \(y^2\). Here it is the number, 5. Remember:

• The quadratic term is always first in the expression.
• The quadratic coefficient shows us the "weight" of the \(y^2\) term.

Understanding the quadratic coefficient is important because it influences the behavior of the quadratic curve when graphed. It's also a building block in the factoring process.This coefficient plays a key role when we calculate the product \(ac\) necessary for further steps. It tells us how much we need to "stretch" or "compress" the squared term to fit the equation.

Factor by Grouping

Factor by grouping is a useful method for solving quadratic expressions when simple factoring isn't straightforward. This technique involves grouping terms to simplify the expression, making it easier to find a pair of factors.Here's how it works in our example:

• We break down the expression \(5y^{2} - 18y - 8\) into two smaller groups: \((5y^{2} - 20y)\) and \((2y - 8)\).
• This process sets us up to identify the greatest common factor (GCF) in each subgroup.
• After factoring out the GCF from each subgroup, we bring the expression back together under a common binomial.

By using factor by grouping, the problem is split into smaller, easier-to-solve chunks, making the solution process more manageable.

Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is the largest factor that divides two or more numbers. Finding the GCF is crucial in simplifying expressions, particularly during factor by grouping.In the expression \(5y^{2} - 18y - 8\), we used GCF twice:

• From \(5y^{2} - 20y\), the GCF is \(5y\).
• From \(2y - 8\), the GCF is \(2\).

After extracting these factors,

• The expression simplifies to \(5y(y - 4) + 2(y - 4)\).

Finding and using the GCF effectively reduces complexity, leading to the exposing common binomials handy for further factoring. This step is subtle yet powerful in transforming a complex expression into a simpler form.

Multiply and Add Method

The multiply and add method is a strategy used to facilitate factoring by finding two magic numbers that fit specific criteria. In our example, this was employed to dissect the quadratic equation.For the expression \(5y^{2} - 18y - 8\):

• First, calculate the product \(ac\): \(5 \times -8 = -40\).
• Find two numbers that multiply to \(-40\) and add up to \(-18\), which are \(-20\) and \(2\).

These numbers allow rearranging the middle term \(-18y\) into \(-20y + 2y\), setting up for factor by grouping.This method complements factor by grouping by identifying key numbers that fit both the multiplication and addition criteria, paving the way for a successful factorization.