Problem 94 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$-4 x(3 x-5)^{2}$$

Step-by-Step Solution

Answer

The result is \(-36x^3 + 120x^2 - 100x\).

Step 1: Recognize the Expression

We'll start with the expression \(-4x(3x - 5)^2\). This expression consists of a monomial \(-4x\) and a binomial \((3x - 5)^2\). Our goal is to multiply them together.

Step 2: Expand the Squared Binomial

Next, we need to expand the term \((3x - 5)^2\). Using the formula for the square of a binomial, \((a - b)^2 = a^2 - 2ab + b^2\), we get: \((3x - 5)^2 = (3x)^2 - 2(3x)(5) + 5^2\). Calculating this gives us:\[(3x)^2 = 9x^2\] \[-2(3x)(5) = -30x\] \[5^2 = 25\] So, \((3x - 5)^2 = 9x^2 - 30x + 25\).

Step 3: Distribute the Monomial

We now distribute the monomial \(-4x\) across each term of the expanded binomial \(9x^2 - 30x + 25\). You do this by multiplying \(-4x\) with each term:1. \(-4x \times 9x^2 = -36x^3\)2. \(-4x \times -30x = 120x^2\)3. \(-4x \times 25 = -100x\).

Step 4: Combine All Terms

Finally, we combine all our terms from the distribution:\[-36x^3 + 120x^2 - 100x\].This is the fully expanded and simplified form of the given expression.

Key Concepts

Binomial Expansion

The binomial expansion refers to taking an expression of the form \((a - b)^2\) and rewriting it in an expanded form using algebraic identities. This involves using the formula \((a - b)^2 = a^2 - 2ab + b^2\), which helps to break it down into separate terms. By applying this formula, we simplify the exponentiated binomial into individual monomials which are easier to handle.
For instance, when expanding \((3x - 5)^2\), each term is calculated separately:

• \((3x)^2\) becomes \(9x^2\)
• \(-2(3x)(5)\) simplifies to \(-30x\)
• \(5^2\) becomes \(25\)

The final product of these expansions gives the expression \(9x^2 - 30x + 25\). Knowing this method is crucial for handling complex polynomial operations effectively.

Polynomial Multiplication

Polynomial multiplication involves the multiplication of expressions containing multiple terms. The crucial part of this process is ensuring that each term in one polynomial is multiplied by each term in the other polynomial.
In our exercise, after expanding the binomial, we multiplied the resulting polynomial \(9x^2 - 30x + 25\) with the monomial \(-4x\). The process goes term by term:

• Multiplying \(-4x\) by \(9x^2\) results in \(-36x^3\)
• The term \(-4x\) multiplied by \(-30x\) gives \(120x^2\)
• Lastly, \(-4x\) multiplied by \(25\) yields \(-100x\)

This procedure ensures that each term is appropriately scaled and accounted for, providing the expanded polynomial \(-36x^3 + 120x^2 - 100x\). Understanding this method is foundational for more complex algebraic manipulations.

Distributive Property

The distributive property is an essential principle in algebra used for multiplying a single term by a polynomial. It states that \(a(b + c) = ab + ac\). Here, every term inside the parenthesis is multiplied by the term outside the parenthesis.
In our exercise, we applied this by distributing the monomial \(-4x\) through the expanded binomial \(9x^2 - 30x + 25\). This means:

• The distribution of \(-4x\) to \(9x^2\) leading to \(-36x^3\)
• Applying \(-4x\) to \(-30x\) gives \(120x^2\)
• Finally, distributing \(-4x\) to \(25\) results in \(-100x\)

The result after using the distributive property is a fully expanded polynomial expression \(-36x^3 + 120x^2 - 100x\). This property is fundamental for simplifying and solving algebraic expressions efficiently.