Question

Multiply the polynomials using the FOIL method. Express your answer as a single polynomial in standard form. $$ (x-2 y)(x+y) $$

Short answer

\(x^2 - xy - 2y^2\)

Step-by-step solution

First Terms

Multiply the first terms in each binomial: The first terms are \(x\) and \(x\). So, \(x \times x = x^2\).

Outer Terms

Multiply the outer terms in each binomial: The outer terms are \(x\) and \(y\). So, \(x \times y = xy\).

Inner Terms

Multiply the inner terms in each binomial: The inner terms are \(-2y\) and \(x\). So, \(-2y \times x = -2xy\).

Last Terms

Multiply the last terms in each binomial: The last terms are \(-2y\) and \(y\). So, \(-2y \times y = -2y^2\).

Combine Like Terms

Combine all the products obtained from the previous steps: \(x^2 + xy - 2xy - 2y^2\). Simplify by combining the like terms: \(x^2 - xy - 2y^2\).

Write in Standard Form

Ensure that the polynomial is in standard form, which orders terms by descending powers of \(x\). The final polynomial in standard form is: \(x^2 - xy - 2y^2\).

Key concepts

Polynomial Multiplication

When multiplying polynomials, we're essentially carrying out multiple multiplications of terms and then adding the results. For the binomials \((x - 2y)(x + y)\), we use the FOIL method. FOIL stands for First, Outer, Inner, Last, referring to the order in which we multiply the terms of each binomial. Let's break it down further:

• **First Terms**: Multiply the first terms in each binomial. Here, it’s \((x \times x = x^2)\).
• **Outer Terms**: Multiply the outer terms. Here, it’s \((x \times y = xy)\).
• **Inner Terms**: Multiply the inner terms. Here, it’s \((-2y \times x = -2xy)\).
• **Last Terms**: Multiply the last terms in each binomial. Here, it’s \((-2y \times y = -2y^2)\).

Performing these multiplications individually gives us \(x^2, xy, -2xy, -2y^2\). Next, we'll combine like terms.

Combining Like Terms

After using the FOIL method, you'll often end up with several similar terms that can be combined. Like terms are terms that have the exact same variables raised to the exact same powers. In our example, we get the set of terms: \(x^2, xy, -2xy, -2y^2\).

• Combine the terms with \(xy\): \((xy - 2xy = -xy)\).
• Add this result to the other terms. This gives us: \((x^2 - xy - 2y^2)\).

Combining like terms helps to simplify the polynomial and makes it easier to work with in future problems.

Standard Form of a Polynomial

Once our polynomial is simplified, we need to write it in standard form. The standard form of a polynomial arranges terms by decreasing powers of the variable. For a polynomial in two variables, arrange by one variable first, usually the one with the highest exponent. In our problem, the final polynomial is \(x^2 - xy - 2y^2\).

• Ensure terms are ordered by descending powers of \(x\).
• If terms have the same power of \(x\), the term with higher \(y\) power comes next.

Writing polynomials in standard form is crucial for comparison and solving equations. It offers clarity and uniformity.