Problem 16 - College Algebra

Question

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

The given polynomial is \( x^2 (2x-3)^2 \). First, we need to expand \((2x-3)^2\). Applying the formula for the square of a binomial, \((a-b)^2 = a^2 - 2ab + b^2\), we get: \((2x-3)^2 = (2x)^2 - 2 \cdot 2x \cdot 3 + 3^2 = 4x^2 - 12x + 9\).

Step 2: Multiply by the remaining term

Now, we multiply \(x^2\) by the expanded form of \((2x-3)^2\), which is \(4x^2 - 12x + 9\). Distribute \(x^2\) to each term in the polynomial: \(x^2 \cdot 4x^2 = 4x^4\), \(x^2 \cdot (-12x) = -12x^3\), and \(x^2 \cdot 9 = 9x^2\). Combine these terms: \(4x^4 - 12x^3 + 9x^2\).

Step 3: Identify the degree of the polynomial

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

Step 4: Determine the leading coefficient

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

Key Concepts

Degree of a Polynomial

When working with polynomials, the **degree** is a key concept. It simply refers to the highest power of the variable present in a polynomial expression. To find the degree, you look for the term that contains the variable with the largest exponent. In the given polynomial \[4x^4 - 12x^3 + 9x^2\],the highest power of \(x\) is 4, which appears in the term \(4x^4\). This means the degree of this polynomial is 4.Knowing the degree of a polynomial is really useful because:

• It helps you understand the maximum number of roots (or solutions) that the polynomial can have.
• It indicates the polynomial's behavior at extreme values of the variable; for example, a degree \(n\) polynomial can have up to \(n-1\) turning points in its graph.

So, always start by identifying that highest exponent—it tells you a lot about the polynomial's nature.

Leading Coefficient

Once you've identified the polynomial's degree, the next crucial piece of information is the **leading coefficient**. This is the coefficient attached to the term with the highest degree. In \[4x^4 - 12x^3 + 9x^2\],the leading term is \(4x^4\). Therefore, the leading coefficient is 4.Why is the leading coefficient important?

• The leading coefficient reveals how steeply or flatly the polynomial behaves as you move away from the center towards the extremes for very large or small values of the variable.
• It directly influences the end behavior of the polynomial's graph, determining whether the graph rises or falls as \(x\) approaches positive or negative infinity.

So, while assessing a polynomial, never overlook this coefficient, as it plays a vital role in understanding and interpreting the polynomial's properties.

Binomial Expansion

**Binomial expansion** is a process where you expand expressions of the form \((a+b)^n\) or \((a-b)^n\) into a polynomial. This involves using algebraic formulas like the binomial theorem or specific identities to transform compact expressions into an expanded form.In the exercise given, part of the unraveling required expanding \((2x-3)^2\) using the identity:\[(a-b)^2 = a^2 - 2ab + b^2\]This gives:\[(2x-3)^2 = 4x^2 - 12x + 9\].This step is crucial because it simplifies complex expressions and brings them to a more usable form. Key benefits of binomial expansion include:

• It turns algebraic expressions into easier problems to solve or manage.
• It's instrumental in calculus, where expansions are used for approximation techniques.

Remember, mastering binomial expansion enhances your capability to tackle various mathematical challenges efficiently.