Problem 73 - Algebra A Combined Function

Question

Multiply. $$ (5 y-4)(3 y-1) $$

Step-by-Step Solution

Answer

The result is \(15y^2 - 17y + 4\).

Step 1: Apply the Distributive Property

To multiply two binomials, we apply the distributive property, also known as the FOIL method. FOIL stands for First, Outer, Inner, and Last, which refers to multiplying each component of the binomials with each other. Let's start by multiplying the First terms of each binomial: \((5y) \times (3y) = 15y^2\).

Step 2: Multiply the Outer Terms

Next, we'll multiply the Outer terms from each binomial. These are the first term of the first binomial and the second term of the second binomial: \((5y) \times (-1) = -5y\).

Step 3: Multiply the Inner Terms

Now, multiply the Inner terms from each binomial. These are the second term of the first binomial and the first term of the second binomial: \((-4) \times (3y) = -12y\).

Step 4: Multiply the Last Terms

Finally, multiply the Last terms from each binomial: \((-4) \times (-1) = 4\).

Step 5: Combine Like Terms

Now, combine all the terms we obtained after multiplying: \[15y^2 - 5y - 12y + 4\].Combine the terms with the same variable powers: \[15y^2 - 17y + 4\].

Key Concepts

Distributive Property in Algebra

In algebra, the distributive property is a key strategy used to simplify expressions and solve equations. The property tells us how to multiply a sum or difference by another term, ensuring that each term in the sum or difference is multiplied separately. For example, if you have an expression like \(a(b + c)\), the distributive property states you can distribute \(a\) to both \(b\) and \(c\) in the parentheses, resulting in \(ab + ac\).

This concept is very useful when dealing with binomials and polynomials. When multiplying two binomials, as in our exercise \((5y - 4)(3y - 1)\), we apply the distributive property to ensure each term of the first binomial multiplies with every term of the second binomial. This systematic process is what the FOIL method helps us organize effectively.

Understanding Binomials

Binomials are algebraic expressions that consist of two terms. An example of a binomial is \(5y - 4\). The binomials are important building blocks in algebra and are particularly useful in operations like addition, subtraction, and especially multiplication.

Each term in a binomial can be a number, a variable, or a combination of both, such as \(ax + b\). When working with binomials, it's crucial to recognize their structure, as this aids in using the FOIL method for their multiplication. The FOIL method is specifically structured to handle binomial multiplication efficiently by dividing the process into four distinct products: First, Outer, Inner, and Last. This allows for systematic distribution of every term, ensuring no component is left unmultiplied.

The Process of Polynomial Multiplication

When multiplying polynomials, like the binomials in our example \((5y - 4)(3y - 1)\), you are essentially applying the distributive property repeatedly. Polynomial multiplication demands that every term in one polynomial must be multiplied by every term in the other polynomial.

The FOIL method is a shortcut technique specifically for binomial multiplication, but for any polynomials beyond two terms, you need to ensure you systematically distribute all terms. Once the multiplication is complete, the next step is to combine like terms. "Like terms" are those that have the same variable raised to the same power. In our exercise, after applying the FOIL method, you would get: \(15y^2 - 5y - 12y + 4\).

• The first product yields \(15y^2\).
• The outer and inner products give \(-5y\) and \(-12y\), respectively.
• The last product gives \(+4\).

By combining these like terms \(-5y\) and \(-12y\), you finally arrive at \(15y^2 - 17y + 4\). These processes solidify your understanding of polynomial multiplication and prepare you for more complex algebraic expressions.