Problem 74 - Algebra A Combined Function

Question

Multiply. $$ (4 z-7)(7 z-1) $$

Step-by-Step Solution

Answer

The product is \(28z^2 - 53z + 7\).

Step 1: Review the Distribution Method

To multiply two binomials, we use the distributive property, often called the FOIL method (First, Outside, Inside, Last). This method involves multiplying each term in the first binomial by each term in the second binomial.

Step 2: Multiply the First Terms

Multiply the first terms of each binomial. Here, multiply \(4z\) by \(7z\). \[4z \times 7z = 28z^2\]

Step 3: Multiply the Outside Terms

Next, multiply the outside terms. Here, multiply \(4z\) by \(-1\). \[4z \times (-1) = -4z\]

Step 4: Multiply the Inside Terms

Now, multiply the inside terms. Here, multiply \(-7\) by \(7z\). \[-7 \times 7z = -49z\]

Step 5: Multiply the Last Terms

Finally, multiply the last terms of each binomial. Multiply \(-7\) by \(-1\). \[-7 \times (-1) = 7\]

Step 6: Combine All Terms

Add all the results from steps 2 to 5 together to obtain the product of the binomials: \[28z^2 - 4z - 49z + 7\]

Step 7: Simplify the Expression

Combine the like terms (\(-4z\) and \(-49z\)): \[28z^2 - 53z + 7\] This gives us the final simplified expression.

Key Concepts

Distributive property

The distributive property is a fundamental mathematical principle that simplifies the multiplication of expressions by spreading terms across parentheses. When dealing with binomials like \[(4z - 7)(7z - 1)\],we use this property to multiply each term in the first binomial by each term in the second. This ensures that no term is left out and every possible interaction is accounted for.

By breaking down the expression, we ensure a smooth and error-free calculation. This step is crucial because each term needs to be multiplied precisely to maintain the expression's integrity. Using the distributive property correctly is the key to working through polynomials and more complex algebraic expressions. Remember:

• Multiply every term in the first binomial by every term in the second binomial.
• Record each result to combine them later.

FOIL method

The FOIL method is a specific application of the distributive property, making it easier to remember the steps needed for multiplying two binomials. FOIL stands for:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

This method provides a structured way to ensure that each component of the binomials is considered, which simplifies keeping track of terms. In our example, each component is calculated:

• First: \(4z \times 7z = 28z^2\)
• Outside: \(4z \times (-1) = -4z\)
• Inside: \((-7) \times 7z = -49z\)
• Last: \((-7) \times (-1) = 7\)

Applying FOIL efficiently breaks down what seems complex into manageable steps, which is perfect for learning or double-checking your work quickly.

Combining like terms

Combining like terms is the final and often most straightforward step in simplifying polynomial expressions. Once all terms are laid out from using the distributive property or FOIL method, the focus shifts to identifying and summing similar terms—those that have identical variables with the same powers.

In the example \(28z^2 - 4z - 49z + 7\), observe that both \(-4z\) and \(-49z\) are like terms, meaning they can be combined to simplify the expression. After combining them, the expression simplifies to:\[28z^2 - 53z + 7\]
This step reduces the polynomial to its simplest form, making it easier to interpret and utilize in further equations or problems. Remember:

• Identify terms with the same variable and power.
• Combine the coefficients of these terms.
• Rewrite the simplified expression clearly.

This practice helps streamline calculations and reinforces the understanding of algebraic expressions.