Problem 44 - College Algebra

Question

Find each product. $$(3 x+2)^{2}$$

Step-by-Step Solution

Answer

The product of the binomial \(3x+2\) squared is \(9x^{2}+12x+4\).

Step 1: Identify the Terms of the Binomial

In the binomial \(3x+2\), the terms are \(3x\) (our first term that aligned with \(a\)) and \(2\) (our second term that aligned with \(b\)).

Step 2: Apply the Binomial Square Formula

Next, the formula for the square of a binomial \((a+b)^{2}=a^{2}+2ab+b^{2}\) can be applied. This gives us \((3x)^{2}+2cdot3xcdot2+2^{2}\).

Step 3: Compute the Expression

Now, perform the calculations in the expression obtained in Step 2. The square of \(3x\) is \(9x^{2}\), the product \(2cdot3xcdot2\) is \(12x\), and the square of \(2\) is \(4\). Adding all these together, the result is calculated to be \(9x^{2}+12x+4\).

Key Concepts

Polynomials

Polynomials are fundamental algebraic expressions that consist of variables and coefficients, structured as a sum of terms, each involving a variable raised to a non-negative integer power. For example, in the expression \(3x^2 + 2x + 5\), each term (\(3x^2\), \(2x\), and \(5\)) is a part of the polynomial.

• A monomial is a polynomial with just one term, like \(5x\) or \(-3\).
• A binomial consists of two terms, such as \(3x + 2\). This is what we often see in binomial theorem problems.
• A trinomial has three terms, such as \(x^2 + 2x + 1\).

As you study polynomials, it's beneficial to recognize their terms and degrees. Understanding polynomials will help in manipulating algebraic expressions and solving various mathematical problems.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, operators (like addition and multiplication), and variables. They are the building blocks of algebra, providing a way to represent real-world situations.In the context of the algebraic expression \((3x + 2)^2\), we see how variables interact with coefficients and constants. Applying operations such as addition, multiplication, and exponentiation allows us to transform and simplify algebraic expressions.

• Variables in expressions represent unknown or changeable values, such as \(x\) in \(3x + 2\).
• Coefficients, like \(3\) in \(3x\), are numbers that multiply a variable. They show how many times the variable is being taken into account.
• Constants are standalone numbers in the expression, such as \(2\) in \(3x + 2\).

Understanding how these elements interact makes it easier to solve expressions involving exponents, like evaluating \((3x + 2)^2\) through expansion.

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power, specifically binomials. A classic and highly useful example is when we have an expression like \((a + b)^n\), which can be expanded using the binomial theorem.The binomial theorem tells us that \((a + b)^2 = a^2 + 2ab + b^2\), which is exactly what we apply when expanding \((3x + 2)^2\). Here's how it works:

• Identify the terms \(a\) and \(b\). For \((3x + 2)\), \(a\) is \(3x\) and \(b\) is \(2\).
• Substitute these into the binomial square formula \((a+b)^2 = a^2 + 2ab + b^2\).
• Calculate each part: \(a^2 = (3x)^2 = 9x^2\), \(2ab = 2 \times 3x \times 2 = 12x\), and \(b^2 = 2^2 = 4\).

Adding these together, we achieve the expanded result: \(9x^2 + 12x + 4\). Binomial expansion simplifies problems involving exponents and is a vital skill for tackling higher-level algebraic challenges.