Problem 168 - Precalculus

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 Binomial

We start by expanding the expression \((2x-3)^2\) using the formula \[(a-b)^2 = a^2 - 2ab + b^2\] for \(a = 2x\) and \(b = 3\). This gives us:\[(2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9\]

Step 2: Distribute \(x^2\)

Next, distribute \(x^2\) across each term in the expanded binomial \(4x^2 - 12x + 9\). This multiplication results in:\[x^2(4x^2) + x^2(-12x) + x^2(9) = 4x^4 - 12x^3 + 9x^2\]

Step 3: Identify the Leading Term

The leading term of the polynomial \(4x^4 - 12x^3 + 9x^2\) is the term with the highest exponent. In this case, it is \(4x^4\).

Step 4: Determine the Degree and Leading Coefficient

Since the highest exponent in the polynomial is 4, the degree of the polynomial is 4. The coefficient of the leading term, \(4x^4\), is 4. Hence, the leading coefficient is 4.

Key Concepts

Binomial Expansion

Binomial expansion is a method of expanding expressions that are raised to a power, such as \((a - b)^n\). It's essential when dealing with polynomials like in the original exercise. The binomial theorem or formula provides a systematic way of expanding these expressions without manually multiplying them out each time.

• In this formula, terms are created by multiplying the binomial's terms and then summing the resulting expressions.
• The formula expands \((a - b)^2\) as \(a^2 - 2ab + b^2\).
• It systematically uses coefficients known as binomial coefficients, seen in patterns like Pascal’s Triangle for quick reference in larger expansions.

The exercise uses binomial expansion to transform \((2x - 3)^2\) into \(4x^2 - 12x + 9\), providing simpler terms to work with.

Leading Coefficient

The leading coefficient is a crucial part of a polynomial which helps us understand several attributes of the equation.

• This coefficient is found in the term with the highest power of the variable in polynomial expressions.
• For example, in the polynomial part \(4x^4 - 12x^3 + 9x^2\), the highest exponent is 4, and the leading coefficient is the number multiplying \(x^4\), which is 4.
• Knowing the leading coefficient helps you determine aspects like the polynomial's end behavior.

In practice, especially dimensionally, the leading coefficient indicates the stretch factor of the polynomial which affects graph shapes and intersections.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and arithmetic operations. Understanding them is fundamental for solving equations and problems in algebra.

• They consist of constants like numbers, coefficients which multiply the variables, and the variables themselves which could be \(x, y\), etc.
• Operations include addition, subtraction, multiplication, and division.
• In a polynomial \(4x^4 - 12x^3 + 9x^2\), each "term" is an individual subexpression like \(4x^4\).

Grasping how to manipulate these expressions—through operations like distribution, factoring, and combining like terms—is key to mastering algebra.