Problem 33 - Algebra 1: Concepts and Skills

Question

Write the square of the binomial as a trinomial. $$ (4 b-3)^{2} $$

Step-by-Step Solution

Answer

The trinomial form of the squared binomial \( (4b - 3)^2 \) is \( 16b^2 - 24b + 9 \).

Step 1: Identify the Binomial Terms

The binomial here is \( (4b - 3) \) where \( a = 4b \) and \( b = 3 \).

Step 2: Apply the Formula

According to the formula \( (a - b)^2 = a^2 - 2ab + b^2 \), substitute \( a = 4b \) and \( b = 3 \) into this equation, which gives \((4b)^2 - 2 * 4b * 3 + 3^2 \).

Step 3: Calculate Each Term

Calculate each term to receive \(16b^2 - 24b + 9 \).

Key Concepts

Binomial Squares

When dealing with binomial squares, you are essentially multiplying a binomial by itself. The task often involves converting such a multiplication into a trinomial. A key formula to remember is:

• For any binomial \[ (a + b)^2 \], the expansion is: \[ a^2 + 2ab + b^2 \].
• Similarly, for \[ (a - b)^2 \], it expands as: \[ a^2 - 2ab + b^2 \].

Notice how the square of a binomial involves three terms, making the result a trinomial. These formulas are handy for quickly expanding binomials, saving time in calculations and solving problems congruently.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. These expressions are foundational in algebra and are used to formulate problems and equations. Consider the expression given:

• The term \[ 4b \] is a part of the binomial, which, upon expansion, is squared to become \[ 16b^2 \].
• The constant term \[ -3 \] expands and contributes to making the combined term \[ +9 \].
• The interaction term \[ -2 imes 4b imes 3 \] simplifies to \[ -24b \].

Algebraic expressions in this form are excellent for practicing substitution and simplifying complex expressions into more manageable parts. Understanding how to handle each component is critical for solving not just binomial squares, but a wide variety of algebraic challenges.

Trinomials

Trinomials consist of three distinct terms. These terms result from the expansion of binomials. In our original task, expanding \[ (4b - 3)^2 \] gives the trinomial \[ 16b^2 - 24b + 9 \]. Here’s how these terms are formed:

• The first term \[ 16b^2 \] comes from squaring the first term of the binomial.
• The second term \[ -24b \] results from twice the product of the binomial's two terms, showing how the cross-term affects the outcome.
• The third term \[ +9 \] is simply the square of the constant term.

Trinomials play a significant role, especially in quadratic equations and factorizations. Understanding how to transform and manipulate them gives you a powerful toolset for solving a broad range of mathematical problems.