Question

The expression \((4 n-5)(5 n-4)\) is equivalent to which of the following? A) \(20 n^2-41 n+20\) B) \(20 n^2-39 n+9\) C) \(9 n^2-41 n+20\) D) \(4 n^2-18 n+9\)

Short answer

The expression \((4 n-5)(5 n-4)\) is equivalent to option A) \(20 n^2-41 n+20\).

Step-by-step solution

Expand the expression

To expand the expression \((4 n-5)(5 n-4)\), we can use the distributive property (also known as the FOIL method) which states that \((a + b)(c + d) = ac + ad + bc + bd\). In our case, a = 4n, b = -5, c = 5n, and d = -4.

Multiply the terms

Using the distributive property, multiply the terms as follows: \[(4n - 5)(5n - 4) = (4n)(5n) + (4n)(-4) + (-5)(5n) + (-5)(-4)\]

Simplify the expanded expression

Perform the operations and simplify: \[20n^2 - 16n - 25n + 20 = 20n^2 - 41n + 20\] The simplified expression is \(20n^2 - 41n + 20\).

Compare with the answer choices

Now, we will compare our simplified expression with the given answer choices: A) \(20 n^2-41 n+20\) ⇒ This matches our simplified expression. B) \(20 n^2-39 n+9\) C) \(9 n^2-41 n+20\) D) \(4 n^2-18 n+9\) The correct equivalent expression for \((4 n-5)(5 n-4)\) is given by option A) \(20 n^2-41 n+20\).

Key concepts

Distributive Property

Understanding the distributive property is essential for algebra students, as it's a fundamental principle used in various mathematical computations. In its simplest form, the distributive property states that multiplying a single term by a group of terms added together is equivalent to doing each multiplication separately. The formula for the distributive property can be written as: \[ a(b + c) = ab + ac \]
In practice, this means that if you have an expression like \(3(x + 4)\), you can distribute the multiplication of 3 across both \(x\) and 4, resulting in \(3x + 12\). When it comes to binomials, the property is often applied through the FOIL method, which stands for 'First, Outer, Inner, Last'. This mnemonic helps remember the order in which to multiply the terms when using the property to expand expressions such as \(4n-5)(5n-4)\).

To expand such binomials, we multiply the First terms, then the Outer terms, the Inner terms, and finally the Last terms, adding the results together to get the expanded form.

Expanding Algebraic Expressions

The Nuts and Bolts of Expansion

Expanding algebraic expressions using the FOIL method allows us to transform a product of binomials into a polynomial, which makes it easier to perform further algebraic manipulations like simplifying or solving equations. Expanding expressions lays the groundwork for understanding more complex algebraic operations and is a skill that benefits from practice.

For instance, given the expression \(4n-5)(5n-4)\), expanding it requires us to distribute each term in the first binomial across each term in the second binomial. This is done systematically through the FOIL method to ensure all terms are accounted for. The product of the First terms \(4n\) and \(5n\) is \(20n^2\), and so on for the Outer, Inner, and Last terms, eventually providing a full polynomial expression.

Simplifying Algebraic Expressions

Streamlining Expressions

Simplifying algebraic expressions involves reducing them to their simplest form by combining like terms and performing any arithmetic operations. Like terms are terms that contain the same variables to the same power. After expanding expressions, as done with the FOIL method, simplification makes expressions more concise and manageable.

Take the expanded result of our previous example, \(20n^2 - 16n - 25n + 20\). The terms \( -16n\) and \( -25n\) are like terms, so we combine them to get \( -41n\), resulting in a simplified form of \(20n^2 - 41n + 20\). This version of the expression is easier to evaluate or utilize in further equations and is the answer that corresponds to the options provided in our original exercise.