Problem 62 - College Algebra with Modeling and Visualization

Question

Multiply the binomials. $$\left(z-\frac{1}{3}\right)\left(z-\frac{1}{6}\right)$$

Step-by-Step Solution

Answer

The product is \( z^2 - \frac{1}{2}z + \frac{1}{18} \).

Step 1: Use the FOIL Method

We will multiply the binomials using the FOIL (First, Outer, Inner, Last) method. Each letter represents the position of the terms in each binomial that we multiply together. Let's start by identifying each pair: First: \( z \cdot z \), Outer: \( z \cdot -\frac{1}{6} \), Inner: \( -\frac{1}{3} \cdot z \), Last: \( -\frac{1}{3} \cdot -\frac{1}{6} \).

Step 2: Multiply the 'First' Terms

Multiply the first terms: \( z \cdot z = z^2 \). This is the first part of our resulting quadratic expression.

Step 3: Multiply the 'Outer' Terms

Multiply the outer terms: \( z \cdot -\frac{1}{6} = -\frac{1}{6}z \). This adds to the linear coefficient of our expression.

Step 4: Multiply the 'Inner' Terms

Multiply the inner terms: \( -\frac{1}{3} \cdot z = -\frac{1}{3}z \). This will also add to our linear coefficient.

Step 5: Multiply the 'Last' Terms

Multiply the last terms: \( -\frac{1}{3} \cdot -\frac{1}{6} = \frac{1}{18} \). This is the constant term of the resulting expression.

Step 6: Combine Like Terms

Now add all the terms together: \( z^2 - \frac{1}{6}z - \frac{1}{3}z + \frac{1}{18} \). Combine the linear terms: \(-\frac{1}{6}z - \frac{1}{3}z = -\left(rac{1}{6} + \frac{2}{6}\right)z = -\frac{3}{6}z = -\frac{1}{2}z \). The final expression is \( z^2 - \frac{1}{2}z + \frac{1}{18} \).

Key Concepts

FOIL Method

Multiplying two binomials can seem daunting, but thanks to the FOIL method, it becomes manageable and systematic. FOIL is useful for remembering how to distribute two binomials, such as in our example \( \left(z-\frac{1}{3}\right)\left(z-\frac{1}{6}\right) \). Each letter in FOIL stands for the terms that should be multiplied together:

• First: Multiply the first terms in each binomial. Here, it is \( z \times z = z^2 \).
• Outer: Multiply the outer terms of the binomials. This is \( z \times -\frac{1}{6} = -\frac{1}{6}z \).
• Inner: Multiply the inner terms, which are \( -\frac{1}{3} \times z = -\frac{1}{3}z \).
• Last: Multiply the last terms of each binomial: \( -\frac{1}{3} \times -\frac{1}{6} = \frac{1}{18} \).

The FOIL method organizes the process clearly, ensuring all elements are correctly included. After applying FOIL, the next step is to combine any like terms. In this exercise, our combined linear terms are \( -\frac{1}{6}z - \frac{1}{3}z \) which sum to \(-\frac{1}{2}z \).

Polynomial Expansion

When expanding a polynomial, you are essentially unraveling an expression to show it in a single polynomial form. Our task was to expand the binomial product \( \left(z-\frac{1}{3}\right)\left(z-\frac{1}{6}\right) \).The first step in a polynomial expansion is to apply the distributive property, systematically multiplying each term in the first binomial by each term in the second. FOIL, as explained earlier, is a specific strategy for achieving this when dealing with binomials.After multiplication, the expansion continues with combining like terms. This makes the expression simpler and easier to interpret. For example, the expansion of our expression results in a polynomial \( z^2 - \frac{1}{2}z + \frac{1}{18} \) after combining the two linear terms. This systematic approach ensures that every part of the binomial is used effectively to create the expanded polynomial.

Quadratic Expressions

Quadratic expressions result from the multiplication of polynomials where the highest degree of the variable is squared. In this exercise, we multiplied two binomials, which resulted in the quadratic expression \( z^2 - \frac{1}{2}z + \frac{1}{18} \).A quadratic expression typically has the form \( ax^2 + bx + c \), where:

• \( a \) is the coefficient of the quadratic term. In our expression, \( a = 1 \).
• \( b \) is the coefficient of the linear term. Here, \( b = -\frac{1}{2} \).
• \( c \) is the constant term. For us, \( c = \frac{1}{18} \).

Quadratic expressions are foundational in algebra and can be interpreted graphically as parabolas. They highlight the significance of polynomial multiplication and combining like terms. Recognizing and working with quadratic forms is crucial for solving equations and analyzing functions in broader algebraic contexts.