Problem 46 - Algebra 1: Concepts and Skills

Question

$$ (4 x+5)(4 x-3) $$

Step-by-Step Solution

Answer

The expanded form of the expression (4x+5)(4x-3) is \(16x^2 + 8x - 15\)

Step 1: Identify the coefficients and terms

Identify the coefficients and terms in the binomial. Here, \(a = 4x\), \(b = 5\), \(c = 4x\), and \(d = 3\)

Step 2: Apply the binomial theorem

Apply the binomial theorem, which is (a + b)(c - d) = ac + bc - ad - bd. Therefore, substitute the values of a, b, c and d to get: \((4x)(4x) + (5)(4x) - (4x)(3) - (5)(3)\)

Step 3: Simplify the expression

Simplify the expression : \(16x^2 + 20x - 12x - 15\) by combining like terms to get: \(16x^2 + 8x - 15\)

Key Concepts

Coefficients

In the world of algebra, coefficients are the numbers in front of the variables in an expression. In the exercise provided, we have the expression \[ (4x + 5)(4x - 3) \],having several coefficients you need to recognize to solve the binomial. Coefficients tell us how much of a variable we have.* For \(4x\), the coefficient is 4. This means 4 is multiplied by the variable \(x\).* The constant terms 5 and \(-3\) also act like coefficients for the term \(1\), even though we usually refer to them as constants since they're not attached to a variable.Recognizing coefficients is crucial because they give us information about what quantities we're dealing with in various parts of the expression. This will make operations like adding, subtracting, and multiplying expressions easier for you.

Simplifying Expressions

Simplifying expressions makes algebraic equations more manageable by reducing them to their simplest form. In our exercise, we have simplified \[ (4x)(4x) + (5)(4x) - (4x)(3) - (5)(3) \] to \(16x^2 + 20x - 12x - 15\). To simplify:

• First, multiply the terms: \((4x)(4x)\) becomes \(16x^2\).
• Then, \((5)(4x)\) results in \(20x\), and \((-4x)(3)\) gives \(-12x\).
• Lastly, \((-5)(3)\) evaluates to \(-15\).

The goal is to express the equation with as few terms as possible, making it easier to understand and solve. Always work towards combining like terms to further simplify.

Combining Like Terms

Combining like terms is the final phase in simplifying expressions. Like terms are terms that have the same variables raised to the same exponent.Looking at our simplified expression from the exercise:\[ 16x^2 + 20x - 12x - 15 \]* The terms \(20x\) and \(-12x\) are like terms because they both have the variable \(x\).* Combine them by adding their coefficients: \(20 - 12 = 8\), resulting in \(8x\).Thus, our final simplified expression becomes:\[ 16x^2 + 8x - 15 \]By combining like terms, you reduce the complexity of expressions. This makes it easier to understand their structure and any calculations needed as part of problem-solving within algebra.