Problem 8 - Intermediate Algebra

Question

Consider the first step of the division process for \(2 x ^ { 2 } - 1 \longdiv { 4 x ^ { 4 } + 0 x ^ { 3 } + 0 x ^ { 2 } + 0 x - 1 }\) How many times does \(2 x^{2}\) divide \(4 x^{4} ?\)

Step-by-Step Solution

Answer

The term \(2x^2\) divides \(4x^4\) exactly \(2x^2\) times.

Step 1: Identify the Leading Terms

In both the divisor and the dividend, identify the leading terms (the highest degree terms). In the divisor, \(2x^2\) is the leading term, and in the dividend, \(4x^4\) is the leading term.

Step 2: Set Up the Division

To determine how many times the leading term of the divisor goes into the leading term of the dividend, divide the leading term of the dividend by the leading term of the divisor, which is \(\frac{4x^4}{2x^2}\).

Step 3: Perform the Division

Calculate \(\frac{4x^4}{2x^2}\). Divide the coefficients: \(\frac{4}{2} = 2\). Subtract the exponents of the like bases: \(x^{4-2} = x^2\). Therefore, \(\frac{4x^4}{2x^2} = 2x^2\).

Step 4: Interpret the Result

The result, \(2x^2\), indicates how many times the \(2x^2\) divides into \(4x^4\). Therefore, \(2x^2\) divides \(4x^4\) exactly \(2x^2\) times.

Key Concepts

Leading Terms

In polynomial division, as in our exercise, identifying the leading terms is a crucial first step. The leading term is the term that has the highest degree in a polynomial. It's like the star player on a sports team, leading the play. For example, in the divisor of our exercise, the leading term is \(2x^2\), and in the dividend, it's \(4x^4\). Recognizing these leading terms helps us understand where to start the division process. Identifying the leading terms enables us to simplify the polynomial division by focusing initially on the terminators. Once we spot them, the division becomes a matter of dividing these lead players, a process that effectively streamlines the solution.

Degree of Polynomials

The degree of a polynomial is determined by the highest power of its variable. It characterizes how 'large' or 'complex' a polynomial can be in terms of its terms with the greatest exponents. For - The divisor \(2x^2\), the degree is 2.- The dividend \(4x^4\), the degree is 4.Understanding the degree is essential in polynomial division because it guides the division process. The degree of the dividend's leading term is initially divided by the degree of the divisor's leading term. This division helps determine the first step of the solution, essentially providing a roadmap on how far we need to decrease the exponents to simplify the expression.

Coefficients and Exponents

Coefficients and exponents are the numbers that appear in a polynomial's terms, influencing its shape and behavior.- **Coefficients** are the numbers before the variable. In \(4x^4\), 4 is the coefficient. They affect the term's amplitude.- **Exponents** are the powers to which the variable is raised. In \(x^4\), 4 is the exponent. They affect the term's degree and lead the polynomial division process.When dividing polynomials, both coefficients and exponents play pivotal roles. We divide the coefficients—\(\frac{4}{2}\)—to get 2. We also subtract the exponents of like bases—\(x^{4-2}\)—to simplify the expression. This combined process of handling coefficients and exponents is what allows us to perform polynomial division correctly, revealing how many times one polynomial is contained within another.