Question

Multiply. $$(2 a-9 b)^{2}$$

Short answer

The squared expression \((2a - 9b)^2\) simplifies to \(4a^2 - 36ab + 81b^2\).

Step-by-step solution

Write the binomial twice

As described in the problem, squaring a binomial means multiplying the binomial by itself. So, first write the expression \((2a - 9b)^2\) as \((2a - 9b) * (2a - 9b)\).

Multiply using FOIL method

Now use the FOIL (First, Outer, Inner, Last) method. Multiply the first terms from each parentheses, the outer terms, the inner terms, and finally the last terms from each parentheses. This will yield: \((2a * 2a) - (2a * 9b) - (9b * 2a) + (9b * 9b)\). Simplifying this will give \(4a^2 - 18ab - 18ab + 81b^2\).

Combine like terms

After performing the multiplication, combine the like terms to simplify the expression. So, \(-18ab - 18ab\) simplifies to \(-36ab\). Therefore, the final expression will be \(4a^2 - 36ab + 81b^2\).

Key concepts

FOIL Method

When you encounter the task of multiplying two binomials, you can effectively use the FOIL method. This acronym stands for First, Outer, Inner, Last and refers to a technique for expanding binomials.

Let's consider an example with the binomials \(x + y\) and \(x - y\). The FOIL method instructs us to multiply the first terms of each binomial (\(x\cdot x\)), the outer terms (\(x\cdot (-y)\)), the inner terms (\(y\cdot x\)), and finally the last terms of each binomial (\(y\cdot (-y)\)).

The resulting terms are then combined to give the expanded form: \[x^2 - xy + xy - y^2 = x^2 - y^2\].
Notice how the like terms \(xy\) and \( - xy\) cancel each other out. The simplicity of the FOIL method makes binomial multiplication more approachable and reduces the likelihood of small mistakes during the process.

Squaring a Binomial

Squaring a binomial, like \(2a-9b\), entails multiplying the binomial by itself. It's a special case of binomial multiplication and can also be done using the FOIL method.

However, an additional pattern emerges with squaring that can make this process even simpler: \[ (a-b)^2 = a^2 - 2ab + b^2 \].
The middle term, \(2ab\), always results from the sum of the inner and outer products of the FOIL method, essentially doubling the product of the binomial's individual terms. Recognizing this pattern when squaring binomials not only saves you time but also helps in avoiding potential errors during the algebraic expansion.

Combining Like Terms

After expanding binomials, you often end up with an expression that has like terms, which are terms with the same variable raised to the same power.

For instance, in our previous example from squaring the binomial \(2a-9b\), we obtained \(4a^2 - 18ab - 18ab + 81b^2\). Here, \-18ab\ and \-18ab\ are like terms and can be combined to simplify the expression, resulting in \(4a^2 - 36ab + 81b^2\).
The process of combining like terms is a fundamental step in simplifying polynomial expressions. It involves adding or subtracting coefficients of these terms while keeping the variable part unchanged, thereby simplifying the overall mathematical expression.

Polynomial Arithmetic

Polynomials are algebraic expressions that consist of variables and coefficients, structured into terms with varying powers. Arithmetic with polynomials involves operations such as addition, subtraction, multiplication, and sometimes division.

When doing polynomial arithmetic, especially multiplication, it’s important to distribute each term of the first polynomial to every term of the second, often using the FOIL method for binomials or a similar distributive approach for polynomials with more terms. After multiplication, we combine like terms to simplify the expression to its most compact form.
Understanding these operations is crucial for solving more complex algebraic equations and for performing calculus operations later in advanced mathematics studies.