Question

Find the product. $$(-a-2 b)^{2}$$

Short answer

The product is \(a^2 + 4ab + 4b^2\).

Step-by-step solution

- Recognize the formula

Recognize that this problem is a perfect square and can be solved using the formula \((a-b)^2 = a^2 - 2ab + b^2 \). Here, \(a = -a\) and \(b = 2b\).

- Substitute the values into the formula

Substitute the values of \(a\) and \(b\) into the formula. So the equation becomes \( (-a)^2 - 2 \cdot -a \cdot 2b + (2b)^2\).

- Simplify the Expression

Simplify the expression. \( (-a)^2 = a^2\), \(-2 \cdot -a \cdot 2b = 4ab\), and \((2b)^2 = 4b^2\). This gives the expression \(a^2 + 4ab + 4b^2\).

Key concepts

Polynomial Multiplication

When we talk about polynomial multiplication, we are referring to the process of multiplying two expressions that consist of variables raised to integer powers, where the variables may be the same or different. This is a foundational concept in algebra and is often visualized through the area model or by using the distributive property.

For example, when squaring a binomial like \( (-a - 2b)^2 \), you effectively multiply the binomial by itself, which translates to \( (-a - 2b)(-a - 2b) \). Here's a quick rundown on how polynomial multiplication works:

• Every term in the first polynomial gets multiplied by every term in the second polynomial.
• All like terms (terms that contain the same variables to the same power) are then combined—this often involves simplification.

This systematic approach ensures that you consider each possible pair of terms only once, leading to a complete and simplified polynomial.

Algebraic Expressions

Algebraic expressions are the bedrock of algebra and encompass pretty much any combination of numbers, variables, and operations (addition, subtraction, multiplication, division, and exponentiation). An expression doesn't have an equality sign, so it's not an equation, but it can be a part of one.

Understanding how to handle these expressions is crucial. When dealing with expressions like \( (-a - 2b)^2 \), you're not just doing arithmetic; you’re applying the rules of algebra to manipulate and simplify the terms involved. It's key to recognize the structure of the expression—whether it’s a monomial, binomial, or trinomial—to effectively use methods such as FOIL or binomial squares to expand or simplify the expression.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while keeping their value the same. The purpose is to make the expressions easier to read or further manipulate.

Here’s what simplification typically involves:

• Combining like terms, which are terms that have the same variables raised to the same power.
• Using arithmetic operations to simplify coefficients and constants.
• Factoring or expanding expressions, depending on the context.

In the context of our example \( (-a - 2b)^2 \), simplifying the expression after multiplying it out involves squaring the terms and combining any like terms that result from the multiplication. Technically, if the entire expression were additive, we’d add together like terms. Since ours is a perfect square, it simplifies into a format which doesn’t require combining like terms.

Binomial Squares

Binomial squares describe the process of squaring a binomial, which is an algebraic expression that contains exactly two terms, like \( (a+b) \) or \( (-a-2b) \) in our example. When you square a binomial, you're multiplying it by itself.

The resulting expressions are called perfect square trinomials because they always follow the format \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\), depending on the sign between the terms in the original binomial. To square a binomial effectively, remember these steps:

• Square the first term.
• Multiply both terms together and double the product for the middle term.
• Square the second term for the last term of the trinomial.

Using the formula for binomial squares can significantly speed up your algebra work and assist in recognizing patterns that lead to quicker simplification of polynomial expressions.