What is the degree of 2x^4-3x^3+6 and x^3+5x^2-15. How did you determine the degree of the polynomials write a rule for finding the leading coefficient

Answer 1

The degree of a polynomial is defined as the highest power of the variable in that polynomial.

For the given polynomials:

1. Polynomial: \(2x^4 - 3x^3 + 6\)

   • The highest power of \(x\) in this polynomial is 4 (from the term \(2x^4\)).
   • Therefore, the degree of the polynomial \(2x^4 - 3x^3 + 6\) is 4.

2. Polynomial: \(x^3 + 5x^2 - 15\)

   • The highest power of \(x\) in this polynomial is 3 (from the term \(x^3\)).
   • Therefore, the degree of the polynomial \(x^3 + 5x^2 - 15\) is 3.

Rule for Finding the Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree.

To find the leading coefficient:

1. Identify the term with the highest degree in the polynomial.
2. The coefficient of this term is the leading coefficient.

Applying this rule:

• For \(2x^4 - 3x^3 + 6\):

  • The highest degree term is \(2x^4\).
  • The leading coefficient is 2.

• For \(x^3 + 5x^2 - 15\):

  • The highest degree term is \(x^3\) (which has a coefficient of 1).
  • The leading coefficient is 1.

Summary

• Degree of \(2x^4 - 3x^3 + 6\): 4, Leading coefficient: 2
• Degree of \(x^3 + 5x^2 - 15\): 3, Leading coefficient: 1