Problem 112 - Beginning and Intermediate Algebra

Question

Find the product. $$4(y-3)^{2}$$

Step-by-Step Solution

Answer

The short answer for the given expression \(4(y-3)^{2}\) is: \(4y^2 - 24y + 36\).

Step 1: Expand the expression inside the parentheses using the distributive property

We will first expand the expression \((y-3)^{2}\) which means \((y-3)*(y-3)\). By multiplying the terms, we get: \[ (y-3)^{2} = y^2 - 3y - 3y + 9 \]

Step 2: Simplify and combine like terms

After expanding the expression, we combine the like terms to get: \[ (y-3)^{2} = y^2 - 6y + 9 \]

Step 3: Multiply the simplified expression by the term outside the parentheses

Now, we multiply the term 4 to the simplified expression: \[ 4(y^2 - 6y + 9) \]

Step 4: Apply the distributive property and simplify

We multiply each term inside the parentheses by 4 and get: \[ 4y^2 - 24y + 36 \] So, the product of the given expression is: \(4(y-3)^{2} = 4y^2 - 24y + 36\).

Key Concepts

Distributive Property

Understanding the distributive property is essential when you're dealing with polynomial multiplication. This property allows you to multiply a sum or difference by another term outside the parentheses. It states that you multiply the term outside by each term inside the parentheses separately, then add the results together.

For instance, when we have an expression like \(a(b + c)\), the distributive property tells us to distribute \(a\) to both \(b\) and \(c\). Hence, the expression becomes \(ab + ac\).

• Useful when expanding expressions that have been factored.
• Keeps equations balanced and simple.
• Helps in transforming complex multiplication tasks into simpler, seriatim calculations.

Applying this to our problem at hand, we see this in action where the term 4 is multiplied across the final expression \(y^2 - 6y + 9\). Each term inside gets multiplied by 4, leading us to our final expression \(4y^2 - 24y + 36\).

Expanding Binomials

Expanding binomials involves transforming expressions where a binomial is raised to a power. This process requires multiplying the expression by itself as many times as indicated by the exponent. For example, \((y-3)^2\) means \((y-3)\times(y-3)\).

To expand it, you use the distributive property to multiply each term of the first binomial by each term of the second:

• Multiply \(y\) by \(y\) to get \(y^2\).
• Multiply \(y\) by \(-3\) to get \(-3y\).
• Multiply \(-3\) by \(y\) to get another \(-3y\).
• Multiply \(-3\) by \(-3\) to get \(+9\).

After computing these products, the terms are combined to give \(y^2 - 3y - 3y + 9\). Notice here, the importance of consistently applying the distributive property with each pair of terms.

Combining Like Terms

Once you've expanded an expression, you'll often end up with terms that can be simplified through combining like terms. Like terms are terms whose variables (and their exponents) are the same. In our example, after expanding \((y-3)^2\), we obtained the expression \(y^2 - 3y - 3y + 9\).

To simplify this, we combine the like terms:

• The terms \(-3y\) and \(-3y\) are like terms, because they both contain \(y\).
• When combined, they result in \(-6y\).

Thereafter, the expression is reduced to \(y^2 - 6y + 9\). This process simplifies the expression, making it easier to work with. It highlights the relationship between similar terms, leading to a more manageable form which can be utilized in further calculations or applications across different mathematical problems.