To determine the degree of the given polynomial, we need to look at the highest power of the variable [tex]\( x \)[/tex] present in the polynomial. The polynomial given is:
[tex]\[ 6x^6 + 9x^3 + 3x^2 - 4x^{10} - 9x^5 - 5x^6 \][/tex]
Let's simplify the polynomial by combining like terms. Terms with the same power of [tex]\( x \)[/tex] can be combined:
1. Combine the [tex]\( x^6 \)[/tex] terms:
[tex]\[ 6x^6 - 5x^6 = (6 - 5)x^6 = 1x^6 = x^6 \][/tex]
Now the polynomial is:
[tex]\[ x^6 + 9x^3 + 3x^2 - 4x^{10} - 9x^5 \][/tex]
2. List out the terms with their respective powers:
- [tex]\( x^{10} \)[/tex] term: [tex]\(-4x^{10}\)[/tex]
- [tex]\( x^6 \)[/tex] term: [tex]\(x^6\)[/tex]
- [tex]\( x^5 \)[/tex] term: [tex]\(-9x^5\)[/tex]
- [tex]\( x^3 \)[/tex] term: [tex]\(9x^3\)[/tex]
- [tex]\( x^2 \)[/tex] term: [tex]\(3x^2\)[/tex]
To find the degree of the polynomial, identify the term with the highest exponent of [tex]\( x \)[/tex]. In this polynomial, the highest exponent is [tex]\( 10 \)[/tex] from the term [tex]\( -4x^{10} \)[/tex].
Therefore, the degree of the polynomial is [tex]\( 10 \)[/tex].
So the correct answer is:
D. 10
