Problem 77 - Algebra 2

Question

Find each product. $$ (2 w+z)(3 w-5 z) $$

Step-by-Step Solution

Answer

\(6w^2 - 7wz - 5z^2\)

Step 1: Identify the binomials

We start by recognizing that we have two binomials to multiply: \((2w + z)\) and \((3w - 5z)\).

Step 2: Use the Distributive Property (FOIL Method)

The product of two binomials is found using the Distributive Property, applying the FOIL (First, Outside, Inside, Last) method. We'll multiply each term in the first binomial by each term in the second binomial.

Step 3: Calculate the First Terms

Multiply the first terms of each binomial: \[ 2w \times 3w = 6w^2 \]

Step 4: Calculate the Outside Terms

Multiply the outside terms: \[ 2w \times -5z = -10wz \]

Step 5: Calculate the Inside Terms

Multiply the inside terms: \[ z \times 3w = 3wz \]

Step 6: Calculate the Last Terms

Multiply the last terms: \[ z \times -5z = -5z^2 \]

Step 7: Combine Like Terms

Now, we combine all the terms obtained: \[ 6w^2 - 10wz + 3wz - 5z^2 \]Combine the like terms \(-10wz\) and \(3wz\):\[ 6w^2 - 7wz - 5z^2 \]

Step 8: Write the Final Answer

After combining the like terms, the final simplified expression is:\[ 6w^2 - 7wz - 5z^2 \]

Key Concepts

Binomials

A binomial is a type of polynomial that contains exactly two terms. In the exercise given, the expressions \(2w + z\) and \(3w - 5z\) are examples of binomials. Each of these expressions consists of two parts that are not like terms, meaning they cannot be combined. Understanding the structure of binomials is essential before applying other mathematical concepts like the distributive property.

• Binomials can have different variables or powers.
• They are fundamental in forming quadratic equations.
• Recognizing binomials helps in simplifying expressions.

By identifying the two terms in each binomial, you can see the need for methods that simplify their multiplication, like the FOIL method.

Distributive Property

The distributive property is a versatile algebraic rule used often in math for multiplication across addition. It basically allows us to distribute a factor across terms inside a parenthesis. When applied to binomials, we use a specific systematic way called the FOIL method. It stands for First, Outside, Inside, Last, which denotes the terms in each binomial that need to be multiplied.

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

This approach makes it easy to ensure all possible products are accounted for, helping to simplify the expression.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together to form a new polynomial. When multiplying binomials, we are in fact engaging in polynomial multiplication. The process involves each term of one polynomial being multiplied with each term of the other polynomial.
In our case, the FOIL method helps in keeping track of multiplying the terms from two binomials together. After using FOIL, you often need to combine like terms to achieve the simplest form of the polynomial.

• Like terms are those terms with the same variables raised to the same powers.
• The result of multiplying two binomials is typically a quadratic polynomial.
• Combining like terms often simplifies the expression substantially.

Ultimately, mastering polynomial multiplication is key to tackling more advanced algebraic equations.