Problem 18 - Algebra Form and Function

Question

Give the leading term. $$ 13 x^{4}\left(2 x^{2}+1\right) $$

Step-by-Step Solution

Answer

Answer: The leading term of the polynomial expression is \(26x^6\).

Step 1: Identify the highest degree term

In the given expression, we have two terms: \(13x^4\) and \((2x^2 + 1)\). When multiplying these terms together, we should focus on finding the highest degree term in the expanded expression.

Step 2: Multiply the highest degree term in each part

Focus on the highest degree terms in each part, which are \(13x^4\) and \(2x^2\). Multiply them together: \((13x^4)(2x^2)\).

Step 3: Simplify the product

Now we simplify the product of the highest degree terms: \[ (13x^4)(2x^2) = 13 \cdot 2 \cdot x^4 \cdot x^2 = 26x^{(4+2)} = 26x^6 \] So, the leading term of the given polynomial expression is \(26x^6\).

Key Concepts

Leading Term

The leading term of a polynomial expression is one of the most crucial components to understand. It gives insight into the expression's general behavior and degree. The leading term is the term with the highest degree in the polynomial after simplification.

For the expression \(13 x^{4}(2 x^{2}+1)\), it involves multiplication of terms. To find the leading term:
1. Identify the term with the highest exponent in each part. Here, those are \(13x^4\) and \(2x^2\).
2. Focus on these terms as they will dictate the leading term after multiplication.
3. Multiply them: \((13x^4)(2x^2)\) results in \(26x^6\), which is the leading term.

Remember, the highest power of the combined terms dominates the polynomial's behavior at large values of \(x\). This knowledge is very useful for understanding the end behavior of the polynomial.

Degree of a Term

The degree of a term is a simple yet fundamental concept in polynomial expressions. It defines the power to which the variable in the term is raised. Understanding this can guide you to spot the leading term easily.

For example, in the term \(13x^4\):

• The coefficient is 13.
• The variable \(x\) is raised to the power 4, making the degree of this term 4.

When looking at a polynomial expression, each term has its degree, and the term with the highest degree tells us about the expression's degree.

In the expression \(13 x^{4}(2 x^{2}+1)\):

• The degrees of the terms we focus on are 4 and 2.
• When multiplied, as in \(x^4 \cdot x^2\), you add the exponents: \(4 + 2 = 6\).
• This results in a term degree of 6, confirming \(26x^6\) as the leading term.

Understanding the degree of terms helps to predict how the polynomial behaves across different values, especially significant values such as infinity.

Multiplication of Polynomials

Multiplying polynomials can initially seem complicated, but when broken down, it's about applying the distributive property and simplifying.

When multiplying the polynomial \(13 x^{4}(2 x^{2}+1)\):

• Distribute each term of the first polynomial (\(13x^4\)) across each term of the second polynomial (\(2x^2 + 1\)).
• Multiply \(13x^4\) by each term: \(2x^2\) and \(1\).
• Perform the multiplication:
  • \((13x^4)(2x^2) = 26x^6\)
  • \((13x^4)(1) = 13x^4\)

The resulting expression \(26x^6 + 13x^4\) simplifies to show \(26x^6\) as the leading term.

This process teaches that careful application of multiplication and attention to degrees leads to understanding how to find leading terms and the importance of term order in polynomial expressions.