Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (x+y)^{2}(x-y)^{2} $$

Step-by-Step Solution

Answer

\(x^4 - 2x^2y^2 + y^4\).

Step 1: Expand Each Binomial

Start by expanding each binomial separately. For \((x+y)^2\), use the formula \((a+b)^2 = a^2 + 2ab + b^2\). This gives us \((x+y)^2 = x^2 + 2xy + y^2\). Next, expand \((x-y)^2\) similarly to \((x-y)^2 = x^2 - 2xy + y^2\).

Step 2: Multiply the Expanded Binomials

Take the expressions from the Step 1 expansion and multiply them: \[(x^2 + 2xy + y^2) \times (x^2 - 2xy + y^2)\]. Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.

Step 3: Distributive Property Multiplication

Multiply each term in the first polynomial \(x^2 + 2xy + y^2\) by each term in the second polynomial \(x^2 - 2xy + y^2\):- \(x^2 \times x^2 = x^4\)- \(x^2 \times (-2xy) = -2x^3y\)- \(x^2 \times y^2 = x^2y^2\)- \(2xy \times x^2 = 2x^3y\)- \(2xy \times (-2xy) = -4x^2y^2\)- \(2xy \times y^2 = 2xy^3\)- \(y^2 \times x^2 = x^2y^2\)- \(y^2 \times (-2xy) = -2xy^3\)- \(y^2 \times y^2 = y^4\).

Step 4: Combine Like Terms

Combine the like terms from the multiplication of polynomials:- There is one \(x^4\) term: \(x^4\).- The \(-2x^3y\) and \(2x^3y\) cancel each other: \(0\).- The \(x^2y^2\) terms are \(x^2y^2 - 4x^2y^2 + x^2y^2 = -2x^2y^2\).- \(2xy^3 - 2xy^3 = 0\).- The \(y^4\) term remains \(y^4\).

Step 5: Present the Final Polynomial

The final polynomial after combining all like terms is:\[ x^4 - 2x^2y^2 + y^4\].

Key Concepts

Binomial Theorem

The binomial theorem is a powerful mathematical tool for expanding expressions that involve powers of binomials. In simpler terms, a binomial is a polynomial with two terms such as \((x + y)\). When raised to a power, we can expand it without multiplying each term manually.
A common special case of the binomial theorem is

• \((a + b)^2 = a^2 + 2ab + b^2\),
• this is precisely what we used in our exercise to expand \((x+y)^2\).

Similarly, the formula helps us expand \((x-y)^2\) using a slight change to account for the negative sign:

• \((a - b)^2 = a^2 - 2ab + b^2\).

The binomial theorem provides a systematic method to handle such expansions, especially for higher powers.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply one term by a set of terms within a parenthesis. It is typically stated as:

• \(a(b + c) = ab + ac\),

where you "distribute" the outer term across the terms inside the parenthesis.
In our exercise, after expanding each binomial, we need to multiply two trinomial expressions. To do this, we apply the distributive property by multiplying each term in the first trinomial by every term in the second:

• This means each combination of terms is multiplied together, ensuring no term is left out,
• for example: \(x^2\) from \((x^2 + 2xy + y^2)\) is multiplied by \(x^2\), \(-2xy\), and \(y^2\) from \((x^2 - 2xy + y^2)\).

This ensures the complete expansion of the product of the two polynomials.

Like Terms

In algebra, like terms are terms whose variables (and their exponents, in case of multiple terms) are the same. The coefficients of these terms can differ.
For instance, \(3x^2\) and \(-5x^2\) are like terms because they share the same variable \(x\) raised to the same power of 2.
When simplifying expressions, it's crucial to combine like terms to simplify the expression.
In the provided solution, after multiplying the terms of the binomials, we identified and combined like terms:

• terms like \(x^2y^2\) appeared multiple times with different coefficients,
• ensuring accurate addition or subtraction of coefficients,
• resulting in \(-2x^2y^2\) in the final polynomial.

Combining like terms is essential for simplifying complex polynomial expressions.

Exponentiation

Exponentiation is a mathematical operation involving numbers raised to a power. It indicates how many times a number, known as the "base," is multiplied by itself.
For example, \(x^3\) means \(x\) is multiplied by itself three times: \(x \times x \times x\).
In our exercise, we dealt with powers while expanding binomials like \((x+y)^2\) and \((x-y)^2\).
Exponentiation also plays a critical role in multiplying terms in algebraic expressions:

• when two exponents with the same base are multiplied, you add their powers.
• For instance: \(x^2 \times x^3 = x^{2+3} = x^5\).

This property helps to keep track of the degrees of terms during polynomial multiplications, allowing for correct simplification.