Problem 68 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (100+27 x)^{2} $$

Step-by-Step Solution

Answer

The product is \(10000 + 5400x + 729x^2\).

Step 1 - Identify the elements of the binomial

In the expression \((100+27x)^2\), we can identify \(a=100\) and \(b=27x\).

Step 2 - Apply the binomial theorem

For \((a+b)^2\), apply the binomial theorem to get \(a^2 + 2ab + b^2\). So, \((100+27x)^2\) equals to \((100)^2 + 2*100*27x + (27x)^2 \).

Step 3 - Simplify the equation

After simplifying the above equation we get \(10000 + 5400x + 729x^2\).

Key Concepts

Expanding Binomials

Expanding binomials involves transforming a binomial expression that is raised to a power into a polynomial expression. A binomial is simply a mathematical expression with two terms, such as

• \((a + b)^n\) or
• \((100 + 27x)^2\)

The process we use to expand binomials relies on a fundamental principle in algebra known as the binomial theorem.

When dealing with a binomial squared, such as \((a + b)^2\), we apply the simple form of the binomial theorem:

• \((a+b)^2 = a^2 + 2ab + b^2\)

This formula allows us to expand the binomial expression into a quadratic form, which is much easier to work with in subsequent calculations.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. It's important to identify that each term in a polynomial expression follows the rule of combining coefficients and variables through addition, subtraction, and multiplication, but not division by variables.

For example, once we expand the binomial expression \((100 + 27x)^2\), we transform it into a polynomial expression:

• \(10000 + 5400x + 729x^2\)

Here, we have a polynomial with three terms, which includes constant and variable parts, showcasing different power levels of the variable \(x\).

Understanding polynomial expressions is crucial, as it allows us to comprehend how expressions are constructed and manipulated, leading to the simplification of complex problems.

Quadratic Equations

Quadratic equations are a type of polynomial equation characterized by the highest power of the variable being a square. These take the general form \(ax^2 + bx + c = 0\), where "a", "b", and "c" are constants.

In our expanded binomial expression,

• \(10000 + 5400x + 729x^2\)

we see a quadratic form emerging with respect to \(x\). Although not set equal to zero, we see similar structuring where we have terms involving \(x^2\) as well as \(x\), typical of a quadratic equation.

Understanding quadratic equations is key because these equations frequently appear in various fields including physics, engineering, and economics. They can also be solved using various techniques like factoring, completing the square, or applying the quadratic formula, which is an essential skill in mathematics.