Enter the degree of the polynomial below.
$5 x^5+8 x^2+2 x-3 x^9-8 x^4-4 x^5$
A. 8
B. 9
C. 5
D. 6
Answer (1)
Combine like terms in the polynomial: 5 x 5 − 4 x 5 = x 5 .
Rewrite the polynomial: x 5 + 8 x 2 + 2 x − 3 x 9 − 8 x 4 .
Identify the term with the highest power of x: − 3 x 9 .
The degree of the polynomial is the exponent of this term: 9 .
Explanation
Understanding the Problem We are asked to find the degree of the polynomial 5 x 5 + 8 x 2 + 2 x − 3 x 9 − 8 x 4 − 4 x 5 . The degree of a polynomial is the highest power of the variable in the polynomial.
Combining Like Terms First, we combine like terms. We have two terms with x 5 : 5 x 5 and − 4 x 5 . Combining these gives us ( 5 − 4 ) x 5 = 1 x 5 = x 5 . So the polynomial becomes x 5 + 8 x 2 + 2 x − 3 x 9 − 8 x 4 .
Identifying the Highest Power Now we need to identify the term with the highest power of x . The terms are x 5 , 8 x 2 , 2 x , − 3 x 9 , and − 8 x 4 . The exponents are 5, 2, 1, 9, and 4, respectively. The highest exponent is 9.
Determining the Degree The term with the highest power of x is − 3 x 9 . The degree of the polynomial is the exponent of this term, which is 9. Therefore, the degree of the polynomial is 9 .
Examples
Understanding the degree of a polynomial is crucial in many areas, such as physics and engineering, where polynomial functions are used to model various phenomena. For example, the trajectory of a projectile can be modeled using a quadratic polynomial, and knowing the degree helps in predicting its path. In economics, polynomial functions can represent cost and revenue curves, and the degree helps in analyzing the behavior of these curves for optimization purposes. This concept is also fundamental in computer graphics for rendering curves and surfaces.
