Problem 44 - A Graphical Approach to College Algebra

Question

Find each product. $$\left(2 z^{4}-3 y\right)^{2}$$

Step-by-Step Solution

Answer

The expanded product is \(4z^8 - 12z^4y + 9y^2\).

Step 1: Identify the Expression

The given expression is \((2z^4 - 3y)^2\). This is a binomial that is being squared.

Step 2: Apply the Binomial Square Formula

Recall the formula \((a - b)^2 = a^2 - 2ab + b^2\). In our case, \(a = 2z^4\) and \(b = 3y\). We will substitute these values into the formula.

Step 3: Calculate the First Term

Find \(a^2\) by squaring \(2z^4\): \((2z^4)^2 = 4z^8\).

Step 4: Calculate the Middle Term

Calculate \(-2ab\) using the identified values:\(-2(2z^4)(3y) = -12z^4y\).

Step 5: Calculate the Last Term

Find \(b^2\) by squaring \(3y\):\((3y)^2 = 9y^2\).

Step 6: Combine the Terms

Combine the results from the previous steps to express the complete expanded form:\(4z^8 - 12z^4y + 9y^2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They form the foundation of algebra and are essential for understanding more complex mathematical concepts. In our exercise, the expression given is \((2z^4 - 3y)^2\). This includes two terms, \(2z^4\) and \(3y\), which are separated by a subtraction operator.

Here are some key points about algebraic expressions:

• **Variables:** These are symbols such as \(z\) and \(y\) in the expression which stand for unspecified numbers.
• **Constant Coefficients:** These are fixed numbers multiplied by the variables, for instance, the \(2\) in \(2z^4\) and the \(3\) in \(3y\).
• **Terms:** The components separated by addition or subtraction signs. \(2z^4\) and \(3y\) are the two terms in the expression.

Understanding how to manipulate algebraic expressions, such as by expanding binomials, is a critical skill in algebra.

Polynomial Squaring

Polynomial squaring involves finding the square of a polynomial, which means multiplying the polynomial by itself. This process is key in algebra and helps simplify expressions to solve problems more effectively.

The expression \((2z^4 - 3y)^2\) is a perfect example of polynomial squaring. Here's how the process works:

• **Identify the Polynomial:** Recognize the expression inside the square bracket, which is our polynomial \((2z^4 - 3y)\).
• **Implement the Squaring Process:** Use the identity \((a - b)^2 = a^2 - 2ab + b^2\) to expand the expression.
• **Calculate Each Part:** Compute the squares of each term \((a^2\) and \(b^2)\) and find the double product \(-2ab\).
• **Combine the Terms:** Add the results obtained in the previous steps. The final expanded form for the polynomial \((2z^4 - 3y)^2\) is \(4z^8 - 12z^4y + 9y^2\).

Polynomial squaring is fundamental in many algebraic procedures, making it a valuable skill to master for students.

Binomial Theorem

The Binomial Theorem provides a method to expand expressions containing a binomial raised to any power. Although our exercise focuses on squaring the binomial, the theorem extends this process to higher powers as well.

In our exercise, the binomial \((2z^4 - 3y)\) clearly shows how the theorem is applied to square the expression:

• **Identify a Binomial Expression:** A binomial is a polynomial with exactly two terms, like \(a - b\) (in our case, \(a = 2z^4\) and \(b = 3y\)).
• **Use the Binomial Square Formula:** For squares, specifically apply the formula \((a - b)^2 = a^2 - 2ab + b^2\).
• **Check Each Step:** Make sure to meticulously perform each calculation step to ensure accuracy (finding \(a^2\), \(-2ab\), and \(b^2\) were critical steps).

By understanding the Binomial Theorem, students can confidently expand binomials to higher powers, using similar structured approaches. The theorem is both simple and versatile for various mathematical applications.