Problem 57 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$(2 z+y)(3 z-4 y)$$

Step-by-Step Solution

Answer

The product is \(6z^2 - 5yz - 4y^2\).

Step 1: Identify Terms for Multiplication

We have two binomials to multiply: \((2z + y)\) and \((3z - 4y)\). Our task is to expand this product using the distributive property (also known as the FOIL method for binomials).

Step 2: Multiply the First Terms

Multiply the first terms from each binomial: \(2z\) from the first binomial and \(3z\) from the second binomial. So: \[2z \times 3z = 6z^2\]

Step 3: Multiply the Outer Terms

Next, multiply the outer terms: \(2z\) from the first and \(-4y\) from the second. This gives: \[2z \times (-4y) = -8zy\]

Step 4: Multiply the Inner Terms

Now multiply the inner terms: \(y\) from the first and \(3z\) from the second. Thus: \[y \times 3z = 3yz\]

Step 5: Multiply the Last Terms

Finally, multiply the last terms from each binomial: \(y\) and \(-4y\). This results in: \[y \times (-4y) = -4y^2\]

Step 6: Combine Like Terms

Now add all the terms together: \[6z^2 - 8zy + 3yz - 4y^2\] Notice that \(-8zy\) and \(3yz\) are like terms. Combine them: \[6z^2 + (-8zy + 3yz) - 4y^2 = 6z^2 - 5zy - 4y^2\]

Step 7: Simplified Expression

The simplified expression after applying all the operations is: \[6z^2 - 5yz - 4y^2\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle of algebra that allows us to multiply a single term by each term inside a bracket. In the context of binomial multiplication, it ensures that each term in one binomial is multiplied by each term in the other.
Imagine you have to distribute candies equally; similarly, in mathematics, you distribute the multiplication across terms.

• The property is mathematically expressed as: \ a(b + c) = ab + ac \.
• For the exercise \( (2z+y)(3z-4y) \), this means multiplying every term in the first binomial by every term in the second.

When you apply this property step by step, you ensure the calculation of every possible product combination. This thoroughness helps in accurate expansion and avoids errors often made by skipping terms. It's essential to carefully track signs (positive or negative) to get the right result. The distributive property simplifies computations and is crucial for problem-solving involving polynomial expressions.

FOIL Method

The FOIL method is a special application of the distributive property for binomials, where FOIL stands for First, Outer, Inner, Last. This mnemonic helps remember the sequence of multiplications necessary to expand two binomials effectively.

• **First**: Multiply the first terms of each binomial. For \( (2z+y)(3z-4y) \), that is \((2z)\times(3z) = 6z^2\).
• **Outer**: Multiply the outer terms. For our binomials, \((2z)\times(-4y) = -8zy\).
• **Inner**: Multiply the inner terms: \(y\times3z = 3yz\).
• **Last**: Multiply the last terms: \(y\times(-4y) = -4y^2\).

The FOIL method brings a systematic approach to expanding binomial products. By following each step, you ensure that no term is overlooked, resulting in a complete expansion. Once the multiplication is done, the next step is to combine any like terms to simplify the expression further.

Like Terms

In algebra, like terms are terms that have identical variable parts and can therefore be combined to simplify expressions. Understanding and identifying like terms is crucial in simplifying your final expression after expansion using distributive property or the FOIL method.

• Like terms are terms that contain the same variables with the same powers. For example, in \(6z^2 - 8zy + 3yz - 4y^2\), \(-8zy\) and \(3yz\) are like terms as they both contain the variables \(z\) and \(y\) raised to the first power.
• Combining like terms means adding or subtracting their coefficients: \(-8zy + 3yz\) simplifies to \(-5yz\).

Identifying like terms helps in reducing the expression to its simplest form. This simplification makes the expression much easier to interpret and use in further calculations or applications. Always take care to respect the signs in front of each term when combining them.