Problem 91 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The expanded form is \(9x^2 - 12x + 4\).

Step 1: Expand the Expression

The expression \[(2 - 3x)^2\] is in the form of \[(a - b)^2\]. Therefore, we can expand it using the formula:\[(a - b)^2 = a^2 - 2ab + b^2\]. Here, \(a = 2\) and \(b = 3x\).

Step 2: Square the First Term

Square the first term (\(a\)), which is 2. This is calculated as:\[2^2 = 4\].

Step 3: Calculate the Double Product

Calculate the double product of the two terms. The two terms are \(2\) and \(3x\). Multiply them together and then double it:\[2 \times 3x = 6x\] and then:\[-2 \times 6x = -12x\].

Step 4: Square the Second Term

Square the second term (\(b\)), which is \(3x\). This is calculated as:\[(3x)^2 = 9x^2\].

Step 5: Combine All Terms

Combine the results of the previous steps:\[4 - 12x + 9x^2\].Rearrange this to write the expression in standard form as:\[9x^2 - 12x + 4\].

Key Concepts

Polynomial Expansion

Polynomial expansion involves expressing a power of a binomial as a sum of terms. It's an essential skill in algebra that allows you to simplify expressions and solve problems more efficiently. In our example, we start with \[(2-3x)^{2}\]which is a binomial raised to the second power.

• To expand this binomial, you need to convert it into a polynomial, where each term has its own coefficient and power of x.
• The standard approach to achieve this is through expansion formulas like the binomial theorem or specific patterns like the square of a binomial, shown in our example.

Breaking the binomial down into its components helps in understanding how each part contributes to the final expansion.

Binomial Theorem

The binomial theorem provides a formula for expanding powers of binomials, letting you multiply binomials quickly. The general expression for \((a+b)^n\) is represented as a series:\[(a+b)^n = \begin{align*}&= a^n + na^{n-1}b + \frac{n(n-1)}{2!}a^{n-2}b^2 + \ldots + b^n. \end{align*}\]In our problem, \[(2 - 3x)^2\]is expanded using a special case of the binomial theorem.

• We identify the binomial components, \(a = 2\) and \(b = -3x\).
• Applying the formula for \((a-b)^2\); it becomes: \[a^2 - 2ab + b^2.\]
• This converts to the polynomial: \[4 - 12x + 9x^2.\]

Though the binomial theorem extends to any power, it simplifies quick expansions through recognizing patterns.

Quadratic Expressions

Quadratic expressions are algebraic expressions of degree two and have the general form:\[ax^2 + bx + c.\]These expressions are foundational in algebra, describing parabolic curves. In our example, after expanding the binomial we derive the quadratic expression:\[9x^2 - 12x + 4.\]

• The term \(9x^2\) indicates the quadratic component, showing the parabola opens upwards since the coefficient is positive.
• The linear term \(-12x\) affects the slope and positioning, representing a leftward shift.
• The constant \(4\) marks where the parabola intersects the y-axis.

Understanding each part’s role in shaping the graph of a quadratic helps in graphing and solving quadratic equations.