Problem 46 - A Graphical Approach to College Algebra

Question

Find each product. $$[(4 y-1)+z]^{2}$$

Step-by-Step Solution

Answer

The expanded expression is \(16y^2 + 8yz - 8y + z^2 - 2z + 1\).

Step 1: Recognize the Expression Structure

The expression \[[(4y - 1) + z]^2\]is in the form of a binomial square, where we have a binomial a = (4y - 1) \text{ and } b = z.

Step 2: Use the Binomial Square Formula

The square of a binomial \[(a + b)^2\]can be expanded using the formula:\[a^2 + 2ab + b^2\].

Step 3: Substitute and Simplify

Substitute a = (4y - 1) \text{ and } b = zinto the formula from Step 2:\[((4y - 1) + z)^2 = (4y - 1)^2 + 2(4y - 1)z + z^2\].

Step 4: Expand \((4y - 1)^2\)

Expand \[(4y - 1)^2\]using the binomial square formula:\[(4y - 1)^2 = (4y)^2 - 2(4y)(1) + 1^2 = 16y^2 - 8y + 1\].

Step 5: Expand \(2(4y - 1)z\)

Distribute and simplify\[2(4y - 1)z = 2 \cdot 4yz - 2z = 8yz - 2z.\]

Step 6: Combine All Terms

Combine all the expanded terms from Steps 4 and 5 into one expression:\[16y^2 - 8y + 1 + 8yz - 2z + z^2.\]

Step 7: Write the Final Expression

Rearrange and write the final simplified expression:\[16y^2 + 8yz - 8y + z^2 - 2z + 1.\]

Key Concepts

Binomial Expansion

When we see an expression like \(((4y - 1) + z)^2\), it's crucial to recognize that it can be expanded using the Binomial Theorem. The Binomial Theorem provides a way to expand expressions that are raised to a power in the form \((a + b)^n\). Here, we're dealing with a special case of the theorem known as the binomial square formula. The square of a binomial \((a + b)^2\) gives us a neat pattern: \(a^2 + 2ab + b^2\).

This pattern allows us to expand \(((4y - 1) + z)^2\) by identifying \(a = (4y - 1)\) and \(b = z\). We then substitute these into the formula, leading us to the expanded expression \((4y - 1)^2 + 2(4y - 1)z + z^2\). With binomials, this expansion method makes it easy to tackle what might seem like a cumbersome multiplication of terms.

Algebraic Expressions

Algebraic expressions like \((4y - 1) + z\) consist of terms that can include constants, variables, and their combinations through operations such as addition, subtraction, multiplication, and division. Understanding how to manipulate and simplify these expressions is a cornerstone of algebra.

In our problem, we see two main terms: \(4y - 1\) and \(z\). To handle these effectively, we first treat each component separately when using the binomial expansion formula. In running through each step, we keep track of operations involving each variable and constant, ensuring to maintain the structure of the expression. This attention to detail is necessary for clear and correct simplification of the initial, more complicated expression.

Polynomial Simplification

Once a polynomial expression has been expanded, the next key step is simplification. This involves combining like terms and organizing the polynomial into a smoother, more concise format. In our example, after applying the binomial expansion, we end up with terms like \(16y^2 - 8y + 1 + 8yz - 2z + z^2\). Each term here has its unique part based on variables \(y\) and \(z\).

Simplification involves identifying like terms and aligning them accordingly. While \(16y^2\) and \(8yz\) stand separate as there's no direct like term to combine them, \(-8y\) and \(-2z\) retain their individual spots due to the absence of similar terms.

The concise polynomial, \(16y^2 + 8yz - 8y + z^2 - 2z + 1\), is now well-organized, making it easier to interpret and use in subsequent algebraic problems. The process of simplification not only aids in solving but also enhances the understanding of polynomial structure and properties.