Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the polynomials. \((x+y)\left(x^{2}-x y+y^{2}\right)\)

Step-by-Step Solution

Answer

The product is \(x^3 + y^3\).

Step 1: Distribute the First Term

Start by distributing the first term of the first binomial over the trinomial. Multiply \(x\) with each term in \(x^{2} - xy + y^{2}\). This gives: \(x(x^{2}) + x(-xy) + x(y^{2}) = x^{3} - x^{2}y + xy^{2}\).

Step 2: Distribute the Second Term

Now, distribute the second term of the binomial, \(y\), over the trinomial. Multiply \(y\) with each term in \(x^{2} - xy + y^{2}\). This yields: \(y(x^{2}) + y(-xy) + y(y^{2}) = yx^{2} - y^{2}x + y^{3}\).

Step 3: Combine Like Terms

Combine the like terms from the results of Step 1 and Step 2: \(x^{3} - x^{2}y + xy^{2} + yx^{2} - y^{2}x + y^{3}\). First, combine the \(x^{2}y\) terms and the \(xy^{2}\) terms: \(x^{3} + (yx^{2} - x^{2}y) + (xy^{2} - y^{2}x) + y^{3}\). Both \(yx^{2} - x^{2}y\) and \(xy^{2} - y^{2}x\) equal zero, simplifying to: \(x^{3} + y^{3}\).

Step 4: Write the Final Simplified Expression

After combining the like terms, the simplified expression for the product is \(x^{3} + y^{3}\). This represents the full multiplication of the given polynomials.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that lets us multiply a single term by two or more terms within a parenthesis. This property states that for any three numbers, say \( a \), \( b \), and \( c \), the equation \( a(b + c) = ab + ac \)must hold true. This principle allows us to simplify expressions and equations efficiently.
In polynomial multiplication, the distributive property helps us expand expressions like \((x+y)(x^2-xy+y^2)\). Here's how it works:

• First, distribute each term in the first binomial across every term in the trinomial.
• Multiply each combination separately to get individual terms.
• Add them together to form an expanded polynomial.

Using this method, we multiply \( x \) with \( x^2, -xy, \) and \( y^2 \), resulting in \( x^3 - x^2y + xy^2 \). Likewise, distributing \( y \) with each term of the trinomial results in \( yx^2 - y^2x + y^3 \).
The distributive property simplifies complex expressions and is a crucial step in evaluating polynomial products.

Combining Like Terms

After distributing the terms, the next essential step is to combine like terms. Like terms in algebra refers to terms whose variables and their exponents match. For instance, \( 3x^2 \) and \( 4x^2 \) can be combined into \( 7x^2 \) since they share the same variable \( x\) raised to the same power.
In our exercise, upon distributing and multiplying, we acquire terms like \( x^3, -x^2y, xy^2, yx^2, -y^2x, \) and \( y^3 \).
To combine them:

• Identify terms with identical variables and powers, such as \( -x^2y \) and \( yx^2 \).
• Add or subtract their coefficients.
• Cancel out terms that sum to zero.

For example, \( -x^2y + yx^2 = 0 \) and \( xy^2 - y^2x = 0 \). Once all like terms are combined or canceled, we're left with \( x^3 + y^3 \), the simplified form.

Binomials and Trinomials

In algebra, binomials and trinomials are particular types of polynomials with specific numbers of terms. A binomial consists of two terms, while a trinomial consists of three terms.
When multiplying a binomial, such as \((x + y)\), with a trinomial like \((x^2 - xy + y^2)\):

• Each term of the binomial (in this case \(x\) and \(y\)) is multiplied by each term of the trinomial.
• This results in a series of products that are combined using the distributive property.
• Finally, the combined like terms further simplify the expression.

Understanding how binomials and trinomials interact is crucial in algebra, as their multiplication can appear daunting. Nonetheless, by following structured steps like distribution and combination, complex expressions become more manageable.
This clear approach not only simplifies polynomial multiplication but also builds a solid foundation for tackling more advanced algebraic concepts.