Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, find the degree and leading coefficient for the given polynomial. $$ x^{2}(2 x-3)^{2} $$

Step-by-Step Solution

Answer

The degree is 4, and the leading coefficient is 4.

Step 1: Expand the Polynomial

First, we need to expand the given polynomial expression. We have \(x^2(2x-3)^2\). The term \((2x-3)^2\) can be rewritten using the distributive property (also known as FOIL for binomials) as \((2x-3)(2x-3)\).

Step 2: Simplify the Binomial Expansion

Now, expand \((2x-3)(2x-3)\). Apply the distributive property: \[(2x-3)(2x-3) = 2x(2x) + 2x(-3) - 3(2x) - 3(-3)\]This simplifies to:\[4x^2 - 6x - 6x + 9 = 4x^2 - 12x + 9\]

Step 3: Combine the Expanded Terms

Now substitute \((2x-3)^2\) back into the original polynomial:\[x^2(4x^2 - 12x + 9)\]Distribute \(x^2\) to each term inside the parentheses:\[x^2 imes 4x^2 + x^2 imes (-12x) + x^2 imes 9\]This results in:\[4x^4 - 12x^3 + 9x^2\]

Step 4: Determine the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable \(x\) in its expanded form. From the expression \(4x^4 - 12x^3 + 9x^2\), the highest power is 4. Therefore, the degree of this polynomial is 4.

Step 5: Identify the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in the polynomial. For the polynomial \(4x^4 - 12x^3 + 9x^2\), the term with the highest degree is \(4x^4\). Thus, the leading coefficient is 4.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is an important concept to grasp when analyzing these mathematical expressions. Essentially, the degree is determined by the highest power of the variable within the polynomial. For example, in the polynomial expression

• \(4x^4 - 12x^3 + 9x^2\)

we note that the degrees of each term are 4, 3, and 2, respectively. Thus, 4 is the largest degree.
This means the degree of the polynomial is 4.
Understanding the degree helps you learn about the behavior of the polynomial, especially when graphing or solving it. Higher degrees generally involve more complex curves when plotted.
It also becomes critical when you explore further topics such as calculus, where the degree can affect the nature of derivatives.

Leading Coefficient

The leading coefficient is fundamental for describing the leading term of a polynomial, which has significant implications for its overall behavior. It's the coefficient or the number in front of the term with the highest degree.
In the polynomial

• \(4x^4 - 12x^3 + 9x^2\)

we identify the leading term as \(4x^4\).
Here, 4 is called the leading coefficient.
This coefficient is crucial because it influences the end behavior of the polynomial—how the polynomial behaves as the variable tends toward infinity or negative infinity.
For polynomials with higher powers, leading coefficients help determine the direction of the curve whether it opens upward or downward in a graph context.

Binomial Expansion

Binomial expansion is a powerful tool when dealing with expressions in the form of

• \((a+b)^n\)

It refers to expanding expressions raised to a power. In high school and beyond, you'll often encounter problems requiring you to expand binomials, such as

• \((2x-3)^2\)

To expand

• \((2x-3)^2 = (2x-3)(2x-3)\),

you apply distributive property or FOIL method (First, Outer, Inner, Last) to break down into simpler components:

• \(2x \times 2x\)
• \(+ 2x \times (-3)\)
• \(- 3 \times 2x\)
• \(+ (-3) \times (-3)\)

This gives

• \(4x^2 - 6x - 6x + 9 = 4x^2 - 12x + 9\).

The understanding of binomial expansions simplifies complex polynomial multiplication and aids in solving higher degree polynomial equations.

Distributive Property

The distributive property is one of the earliest yet most crucial arithmetic concepts involved in algebra. It states that

• \(a(b + c) = ab + ac\)

This principle allows you to simplify expressions and multiply a single term across a sum or difference inside parentheses.
In the context of our example,

• \(x^2(4x^2 - 12x + 9)\)

applying the distributive property involves multiplying each term inside the parentheses by \(x^2\):

• \(x^2 \times 4x^2 = 4x^4\)
• \(- x^2 \times 12x = -12x^3\)
• \(+ x^2 \times 9 = 9x^2\)

By employing this method, the polynomial expands into its detailed components, allowing you to easily handle and solve polynomial equations. As seen, it simplifies to

• \(4x^4 - 12x^3 + 9x^2\),

easing more complex calculations and helping solve equations systematically.