Certainly! Let's work through each of the given expressions step-by-step to understand their components and structure.
1. Expression 1: [tex]\(3x + x^3\)[/tex]
- This expression contains two terms.
- The first term is [tex]\(3x\)[/tex], which is a linear term. It represents three times the variable [tex]\(x\)[/tex].
- The second term is [tex]\(x^3\)[/tex], which is a cubic term. It represents the variable [tex]\(x\)[/tex] raised to the power of three.
- Therefore, the entire expression is [tex]\(3x + x^3\)[/tex].
2. Expression 2: [tex]\(-4x^2 + 5\)[/tex]
- This expression contains two terms as well.
- The first term is [tex]\(-4x^2\)[/tex], which is a quadratic term. It represents negative four times the variable [tex]\(x\)[/tex] squared.
- The second term is a constant term [tex]\(5\)[/tex].
- Therefore, the entire expression is [tex]\(-4x^2 + 5\)[/tex].
3. Expression 3: [tex]\(-x^3 + 4x^2 - 6\)[/tex]
- This expression contains three terms.
- The first term is [tex]\(-x^3\)[/tex], which is a cubic term. It represents negative one times the variable [tex]\(x\)[/tex] raised to the power of three.
- The second term is [tex]\(4x^2\)[/tex], which is a quadratic term. It represents four times the variable [tex]\(x\)[/tex] squared.
- The third term is a constant term [tex]\(-6\)[/tex].
- Therefore, the entire expression is [tex]\(-x^3 + 4x^2 - 6\)[/tex].
Putting all of this together, the three expressions given are:
[tex]\[3x + x^3\][/tex]
[tex]\[ -4x^2 + 5\][/tex]
[tex]\[-x^3 + 4x^2 - 6\][/tex]
Each expression has its unique components - linear, quadratic, cubic terms, and constants, which define their mathematical structure.
