Problem 76 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (5 y-1)^{2}-(y+7)(y-7) $$

Step-by-Step Solution

Answer

The simplified form is \(24y^{2} - 10y + 50\).

Step 1: Expand the first expression

Apply the formula for the square of a binomial: \[(a-b)^{2} = a^{2} - 2ab + b^{2}\]For our expression \((5y-1)^{2}\), let \(a=5y\) and \(b=1\). Then:\[(5y-1)^{2} = (5y)^{2} - 2(5y)(1) + (1)^{2} = 25y^{2} - 10y + 1\].

Step 2: Expand the second expression

Use the difference of squares formula: \[(a+b)(a-b) = a^{2} - b^{2}\]For the expression \((y+7)(y-7)\), this gives:\[(y+7)(y-7) = y^{2} - 7^2 = y^{2} - 49\].

Step 3: Substitute expanded expressions

Replace the original operations with the expanded forms obtained from the previous steps:\[(5y-1)^{2} - (y+7)(y-7) = (25y^{2} - 10y + 1) - (y^{2} - 49)\].

Step 4: Simplify the expression

Distribute the negative sign across the second expanded expression and combine like terms:\[(25y^{2} - 10y + 1) - y^{2} + 49 = 25y^{2} - y^{2} - 10y + 1 + 49\].Combine like terms:\[24y^{2} - 10y + 50\].

Key Concepts

Binomial Expansion

Binomial expansion is a critical algebraic skill that expands expressions raised to a power. Essentially, it involves using the Binomial Theorem to express powers of sums and differences as a sum of terms. One of the simplest forms is the expansion for a binomial squared:

• The expression \((a-b)^2 = a^2 - 2ab + b^2\) illustrates how this works.
• Here, \(a\) and \(b\) are any two terms.
• This formula helps to break down and fully expand a squared binomial.

In the given exercise, the term \((5y - 1)^2\) was expanded using this formula, where \(5y\) is the first term and \(1\) is the second term. By applying the formula:

• First, calculate \((5y)^2 = 25y^2\).
• Second, compute \(-2 \times 5y \times 1 = -10y\).
• Finally, find \((1)^2 = 1\).

Together, these components give the expansion \(25y^2 - 10y + 1\).
Using this simple formula repeatedly will strengthen your understanding and make handling any squared binomial straightforward.

Difference of Squares

The difference of squares is a special case in algebra that simplifies the multiplication of conjugates. It reflects a unique pattern:

• The formula \((a+b)(a-b) = a^2 - b^2\) captures this.
• It involves two terms squared and subtracted.
• Typically used to simplify multiplications where direct expansion might be cumbersome.

In the provided exercise, the expression \((y+7)(y-7)\) uses this formula directly:

• Here, \(a = y\) and \(b = 7\).
• Thus, \((y+7)(y-7)\) simplifies to \(y^2 - 49\).

This formula not only speeds up calculation but also aids in quickly identifying patterns in expressions involving conjugates. It's especially useful in higher-level algebra where recognizing such structures can save time.

Polynomial Simplification

Polynomial simplification involves combining and reducing polynomial expressions to their simplest form. This process often follows expansion or factorization and involves:

• Distributing operations across terms.
• Combining like terms—terms with the same variable and exponent.
• To "simplify" is to collect all like terms.

In the exercise, once each binomial is expanded, the next step is to simplify the expression:

• The expanded forms are \(25y^2 - 10y + 1\) and \(y^2 - 49\).
• Subtract the second from the first: \[(25y^2 - 10y + 1) - (y^2 - 49)\].
• Distribute the negative sign: \[25y^2 - y^2 - 10y + 49 + 1\].
• Finally, combine like terms to arrive at \(24y^2 - 10y + 50\).

This process results in a simplified polynomial that is both neatly organized and ready for further analysis or solution. Mastering simplification techniques ensures efficiency and accuracy in algebra.