Problem 91 - Algebra and Trigonometry

Question

(a) Show that \(a b=\frac{1}{2}\left[(a+b)^{2}-\left(a^{2}+b^{2}\right)\right] .\) (b) Show that \(\left(a^{2}+b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}=4 a^{2} b^{2}.\) (c) Show that \(\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=(a c+b d)^{2}+(a d-b c)^{2}.\) (d) Factor completely: \(4 a^{2} c^{2}-\left(a^{2}-b^{2}+c^{2}\right)^{2}.\)

Step-by-Step Solution

Answer

(a) Shown by expansion; (b) Shown as difference of squares; (c) Shown by rearranging and expanding; (d) Factor as difference of squares.

Step 1: Expand and Simplify (a)

First, expand the expression \((a + b)^2 - (a^2 + b^2)\). Start with \( (a + b)^2 = a^2 + 2ab + b^2 \). So, the expression becomes: \[(a^2 + 2ab + b^2) - (a^2 + b^2) = 2ab.\] Thus, we have shown that \(ab = \frac{1}{2} \times 2ab \), which confirms \( ab = \frac{1}{2} \left[(a+b)^2 - (a^2+b^2)\right] \).

Step 2: Expand and Simplify (b)

Use the difference of squares property: \((a^2 + b^2)^2 - (a^2 - b^2)^2\) can be written as \[((a^2 + b^2) + (a^2 - b^2))((a^2 + b^2) - (a^2 - b^2)).\] This expands to: \[(2a^2)(2b^2) = 4a^2b^2.\] Therefore, we confirm the equation.

Step 3: Expand and Simplify (c)

First, recognize the expressions on each side. On the left, it's a multiplication of sums \((a^2 + b^2)(c^2 + d^2)\). When expanded, it results in \(a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2\).On the right side, expand \((ac + bd)^2 + (ad - bc)^2\) which is: \[(ac)^2 + 2(ac)(bd) + (bd)^2 + (ad)^2 - 2(ad)(bc) + (bc)^2.\] Combine and rearrange terms to match the left side, confirming the equality.

Step 4: Factor Completely (d)

Given: \(4a^2c^2 - (a^2 - b^2 + c^2)^2\). First, recognize it as a difference of squares:\[(2ac)^2 - (a^2 - b^2 + c^2)^2.\] Let \(u = 2ac\) and \(v = a^2 - b^2 + c^2\). Then, this expression factors as \[(u - v)(u + v).\] Substitute back for \(u\) and \(v\) to factor the expression completely.

Key Concepts

Expansion and Simplification

In algebra, expanding and simplifying expressions is a crucial step to solve equations effectively. Let's consider the expression \((a+b)^2 - (a^2 + b^2)\). Firstly, expand \((a+b)^2\) by applying the distributive property:

• \((a+b)^2 = a^2 + 2ab + b^2\)

Next, subtract \((a^2 + b^2)\) from the expanded expression:

• \(a^2 + 2ab + b^2 - a^2 - b^2 = 2ab\)

This clean-up process, often referred to as simplification, reduces complex expressions to simpler forms. It involves canceling out like terms to make an equation more manageable.

Factorization

Factorization is the process of breaking down an expression into a product of its factors. It’s a method often used to simplify expressions or solve equations. Consider the expression \(4a^2c^2 - (a^2 - b^2 + c^2)^2\). Notice that this takes the form of a difference of squares:

• \((2ac)^2 - (a^2 - b^2 + c^2)^2\)

When you recognize this, rewrite it using the difference of squares identity:

• \((u - v)(u + v)\), where \(u = 2ac\) and \(v = a^2 - b^2 + c^2\)

Factorization helps to simplify complex algebraic expressions and to solve quadratic equations by setting each factor to zero.

Difference of Squares

The difference of squares is a special algebraic identity that helps in quickly factoring expressions. The identity states that \(a^2 - b^2 = (a-b)(a+b)\). It is widely used in simplifying expressions and solving equations. In our context, consider \((a^2 + b^2)^2 - (a^2 - b^2)^2\). Recognize it as the difference of squares:

• \(((a^2 + b^2) + (a^2 - b^2))((a^2 + b^2) - (a^2 - b^2)) = 4a^2b^2\)

This identity makes it straightforward to factor large quadratic terms into simpler expressions. Using difference of squares simplifies calculations and is a fundamental tool in polynomial identities.

Polynomial Identities

Polynomial identities are equations that hold true for all variable values. They are deeply rooted in algebraic manipulation techniques such as expansion and simplification, factorization, and difference of squares. Here, we analyze \((a^2 + b^2)(c^2 + d^2)\) and show its equivalency to \((ac + bd)^2 + (ad - bc)^2\):

• Expanding both expressions, you'll rearrange and combine terms to demonstrate both sides' equality.

These identities help in proving more complex algebraic relationships, reducing polynomial expressions, and solving equations more efficiently. Understanding how to apply these identities aids in comprehending and solving various algebraic problems quickly.