Problem 86 - Algebra and Trigonometry

Question

Perform the indicated operations, and simplify. \((1-b)^{2}(1+b)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(b^4 - 2b^2 + 1\).

Step 1: Expand the First Expression

The given expression is \((1-b)^2(1+b)^2\). Start by expanding \((1-b)^2\) using the formula \((a-b)^2 = a^2 - 2ab + b^2\). Substitute \(a = 1\) and \(b = b\):\[(1-b)^2 = 1^2 - 2 \cdot 1 \cdot b + b^2 = 1 - 2b + b^2\].

Step 2: Expand the Second Expression

Next, expand \((1+b)^2\) using the same formula \((a+b)^2 = a^2 + 2ab + b^2\). Substitute \(a = 1\) and \(b = b\):\[(1+b)^2 = 1^2 + 2 \cdot 1 \cdot b + b^2 = 1 + 2b + b^2\].

Step 3: Multiply the Expanded Expressions

Now multiply the two results from Step 1 and Step 2: \((1 - 2b + b^2)(1 + 2b + b^2)\). Use the distributive property (FOIL method) to expand:1. First by First: \(1 \cdot 1 = 1\)2. Outer by Outer: \(1 \cdot 2b = 2b\)3. Inner by Inner: \(-2b \cdot 1 = -2b\)4. Lasts approach (First term by Last term, Outer by Inner, Inner by Outer, and Lasts: results in simpler steps to visualize): - \(b^2 \cdot b^2 = b^4\) - \(b^2 \cdot 2b = 2b^3\) - \(b^2 \cdot 1 = b^2\) - \(-2b \cdot b^2 = -2b^3\) - \(-2b \cdot 2b = -4b^2\) - \(-2b \cdot 1 = -2b\)5. Last by Last: \(1 \cdot b^2 = b^2\).Combine all these terms:\[1 + 2b - 2b + b^4 + 2b^3 - 2b^3 + b^2 - 4b^2 + b^2 + b^2\].

Step 4: Combine Like Terms

Simplify the resulting expression by combining like terms:\[1 + 2b - 2b + b^4 + (2b^3 - 2b^3) + (b^2 - 4b^2 + b^2 + b^2)\].This simplifies to:- Constant term: \(1\)- Fourth degree term: \(b^4\)- Third degree terms cancel out: \(0\)- Second degree terms: \(-4b^2 + 3b^2 = -2b^2\)- First degree terms cancel out: \(0\)Thus, the simplified expression is:\[b^4 - 2b^2 + 1\].

Key Concepts

Expansion

Expansion is a vital step in polynomial manipulation where terms are spread out to showcase the complete expression. It lets us see the relationships between different parts of a polynomial. For instance, given \((1-b)^2\) or \((1+b)^2\), we use special identities like \((a-b)^2 = a^2 - 2ab + b^2\) or \((a+b)^2 = a^2 + 2ab + b^2\).
- Start with the expression, substitute the required values, and spread out the products.- For example, in expanding \((1-b)^2\), substitute each term appropriately to get \(1 - 2b + b^2\).
Thus, expansion helps uncover each term's contribution within a polynomial. This clarity is crucial for further operations like multiplication or simplifying expressions.

Simplification

After expansion, simplification brings a polynomial to its most manageable form. The aim is to bring together comparable terms, reducing the length of an expression. In our example, expressions like \(1 + 2b - 2b + b^4 + 2b^3 - 2b^3 + b^2 - 4b^2 + b^2 + b^2\) may seem intricate. But by simplifying, you arrive at the compact form:
- Combine terms that have similar properties, such as powers or coefficients.- Cancellation plays a key role. Terms like \(2b\) and \(-2b\) cancel out, simplifying to zero.
Ultimately, practicing simplification transforms complex polynomial expressions into a neater form, making calculations much easier.

Distributive Property

The distributive property is a powerful algebraic tool, essential for multiplying polynomials. It states that \(a(b + c) = ab + ac\). This helps break down expressions into manageable parts. When manually expanding a polynomial like \((1 - 2b + b^2)(1 + 2b + b^2)\), follow the distributive principle. Dividing into steps makes it simpler:
- Multiply each term of one polynomial by every term of the other.- Systematically perform the operations like multiplying "first," "outer," "inner,'' and "last" terms (the FOIL method).
Breaking the expression into parts using the distributive property ensures no term is overlooked. This systematic approach sets the stage for the final simplification stage.

Like Terms

Grouping terms that share the same variable component and power is how we deal with like terms. They refine and simplify polynomial expressions.
- For instance, in a polynomial like \(b^2 - 4b^2 + 3b^2\), you collect all \(b^2\) terms together, resulting in \(-2b^2\).- Like terms only involve similar variables raised to the same power. So, \(b^2\) and \(b^2\) combine, but not \(b^2\) and \(b^3\).
Recognizing and efficiently combining like terms is critical for reducing a polynomial expression to its cleanest form. It's akin to tidying up a room: organizing similar items so the space makes sense.