Problem 42 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (4 x+5)^{2} $$

Step-by-Step Solution

Answer

The product of \((4x+5)^2\) is \(16x^2 + 40x + 25\).

Step 1: Identify the terms of the binomial

The terms of the binomial are \(a = 4x\) and \(b = 5\).

Step 2: Square each term

Square each of the items, so that \(a^2 = (4x)^2 = 16x^2\) and \(b^2 = 5^2 = 25\).

Step 3: Multiply the terms and double the result

Multiply the items (a*b), so \(2ab = 2(4x*5) = 40x\).

Step 4: Sum up all terms

Add up all the terms \(a^2 + 2ab + b^2 \) to get \(16x^2 + 40x + 25\)

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that involve variables raised to whole number powers. They can have one or more terms, each consisting of a coefficient (a constant number) multiplied by a variable raised to an exponent.

In the given exercise, we have a binomial expression, which is a type of polynomial expression with two terms: \(4x + 5\). These terms are separated by either a plus \((+)\) or minus \((-\)) sign. The entire expression is raised to the power of 2, indicating that the binomial is to be multiplied by itself.

• Term 1: \(4x\)
• Term 2: \(5\)
• Exponent: \(2\)

When simplifying polynomial expressions like this, it is essential to recognize the role of each term and how they interact, especially when following operations like squaring or expanding them using special formulas.

Distributive Property

The distributive property is a fundamental principle used to simplify expressions and equations. It involves "distributing" or multiplying a single term across the terms inside parentheses.

In this exercise, to expand \( (4x+5)^2 \), the distributive property plays a crucial role. We first express the binomial squared as multiplying the binomial by itself: \( (4x+5)(4x+5) \). Expanding it requires applying the distributive property to each pair of terms:

• Multiply \(4x\) by each term in the second binomial.
• Do the same by multiplying \(5\) by each term in the second binomial.

This results in: \[ (4x)(4x) + (4x)(5) + (5)(4x) + (5)(5) = 16x^2 + 20x + 20x + 25 \]
Notice that we add the like terms, \(20x + 20x = 40x\), to further simplify the expression to its final expanded form.

Properties of Exponents

The properties of exponents are essential for effectively working with polynomial expressions, particularly when terms are raised to a power.

In the context of our exercise, squaring each term of the binomial relies on the rules governing exponents:

• When raising a monomial \( (ax)^n \) to a power, both the coefficient and the variable are raised to that power. For \( (4x)^2 \), it becomes \(16x^2\).
• The same rule applies to constants, so \(5^2 = 25\).

Understanding these properties allows for simplification and manipulation of expressions. Recognizing how to apply them is crucial for any exercise involving powers and significantly simplifies tasks like polynomial expansion.