Give the degree of each polynomial.
23. [tex]a^3 b^4 c[/tex]
24. [tex]5 f^2 g-3 d^2[/tex]
25. [tex]5 x^4+3 x-14 x^2-x^3[/tex]
26. [tex]20 x^5 y^7-2 w z^9[/tex]
27. 2012
28. 0
29. [tex]\frac{3}{4}[/tex]
30. [tex]-55 g[/tex]
31. [tex]x y z^4-7[/tex]
32. [tex]13 r^2 s-4 t^{12}[/tex]
33. [tex]23 x^4+6 x^8+4 x^2-x^3[/tex]
34. [tex]30 a^{20} b^2[/tex]
Answer (2)
The degree of a 3 b 4 c is 8 .
The degree of 5 f 2 g − 3 d 2 is 3 .
The degree of 5 x 4 + 3 x − 14 x 2 − x 3 is 4 .
The degree of 20 x 5 y 7 − 2 w z 9 is 12 .
The degree of 2012 is 0 .
The degree of 0 is undefined.
The degree of 4 3 is 0 .
The degree of − 55 g is 1 .
The degree of x y z 4 − 7 is 6 .
The degree of 13 r 2 s − 4 t 12 is 12 .
The degree of 23 x 4 + 6 x 8 + 4 x 2 − x 3 is 8 .
The degree of 30 a 20 b 2 is 22 .
Explanation
Understanding Polynomial Degrees We need to find the degree of each polynomial. The degree of a polynomial is the highest degree of its terms. The degree of a term is the sum of the exponents of the variables in that term. A constant term has degree 0. The degree of the zero polynomial is undefined.
Calculating Degree of a^3 b^4 c
a 3 b 4 c : The degree of this term is 3 + 4 + 1 = 8 .
Calculating Degree of 5 f^2 g - 3 d^2
5 f 2 g − 3 d 2 : The degree of the first term is 2 + 1 = 3 . The degree of the second term is 2 . The degree of the polynomial is 3 .
Calculating Degree of 5 x^4 + 3 x - 14 x^2 - x^3
5 x 4 + 3 x − 14 x 2 − x 3 : The degrees of the terms are 4 , 1 , 2 , 3 . The degree of the polynomial is 4 .
Calculating Degree of 20 x^5 y^7 - 2 w z^9
20 x 5 y 7 − 2 w z 9 : The degree of the first term is 5 + 7 = 12 . The degree of the second term is 1 + 9 = 10 . The degree of the polynomial is 12 .
Calculating Degree of 2012
2012 : This is a constant term, so its degree is 0 .
Calculating Degree of 0
0 : This is the zero polynomial, so its degree is undefined.
Calculating Degree of 3/4
4 3 : This is a constant term, so its degree is 0 .
Calculating Degree of -55g
− 55 g : The degree of this term is 1 .
Calculating Degree of x y z^4 - 7
x y z 4 − 7 : The degree of the first term is 1 + 1 + 4 = 6 . The degree of the second term is 0 . The degree of the polynomial is 6 .
Calculating Degree of 13 r^2 s - 4 t^{12}
13 r 2 s − 4 t 12 : The degree of the first term is 2 + 1 = 3 . The degree of the second term is 12 . The degree of the polynomial is 12 .
Calculating Degree of 23 x^4 + 6 x^8 + 4 x^2 - x^3
23 x 4 + 6 x 8 + 4 x 2 − x 3 : The degrees of the terms are 4 , 8 , 2 , 3 . The degree of the polynomial is 8 .
Calculating Degree of 30 a^{20} b^2
30 a 20 b 2 : The degree of this term is 20 + 2 = 22 .
Examples
Understanding the degree of a polynomial is crucial in many areas, such as physics, engineering, and computer graphics. For instance, when modeling the trajectory of a projectile, the equation often involves polynomials. The degree of the polynomial helps determine the complexity of the trajectory. Similarly, in computer graphics, polynomial equations are used to define curves and surfaces, and the degree affects the smoothness and shape of these objects. Knowing the degree helps in choosing appropriate algorithms and computational methods to efficiently solve related problems.
The degrees of the provided polynomials range from 0 to 22, depending on the highest sum of the exponents in each term. Notably, polynomials with only constants have a degree of 0, while the zero polynomial has an undefined degree. The highest degree among the polynomials given is 22 for 30 a 20 b 2 .
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