\(\displaystyle 6x^{5}+5x^{4}-3x^{2}+x^{7}\)

The degree of an individual term of a polynomial is the exponent of its variable; the exponents of the terms of this polynomial are, in order, 5, 4, 2, and 7.

The degree of the polynomial is the highest degree of any of the terms; in this case, it is 7.

\(\displaystyle ab^{2}+a^{2}b^{3}-a^{3}b^{1}\)

To find the degree of the polynomial, you first have to identify each term [term is for example \(\displaystyle ab^{2}\)], so to find the degree of each term you add the exponents.

EX: \(\displaystyle ab^{2}\) \(\displaystyle = a^{1}b^{2} = 1+2= 3\) - Degree of 3

Highest degree is \(\displaystyle 5\rightarrow\) \(\displaystyle a^{2}b^{3}= 2+3= 5\)

To find the degree of the polynomial, add up the exponents of each term and select the highest sum.

12x²y³: 2 + 3 = 5

6xy⁴z: 1 + 4 + 1 = 6

2xz: 1 + 1 = 2

The degree is therefore 6.

When solving functions within functions, we begin with the innermost function and work our way outwards. Therefore:

\(\displaystyle \small h(g(x))=2(x^3)=2x^3\)

and

\(\displaystyle \small f(h(g(x)))=(2x^3)^2-10=4x^6-10\)

This problem relies on our knowledge of a radical expression \(\displaystyle \small \sqrt[b]{x^a}\) equal to \(\displaystyle \small x^\frac{a}{b}\). The functions are subbed into one another in order from most inner to most outer function.

\(\displaystyle \small g(h(x))=(\sqrt[4]{x})^3=x^\frac{3}{4}\)

and

\(\displaystyle \small f(g(h(x)))=\sqrt[3]{(x^\frac{3}{4})^2}=\sqrt[3]{x^\frac{6}{4}}=x^\frac{6}{12}=x^\frac{1}{2}=\sqrt{x}\)

In problems with functions within one another, we must first solve the innermost function and then proceed outwards. Therefore, the first step is solving \(\displaystyle \small g(9)\) :

\(\displaystyle \small g(9) =\small \sqrt{(9)^2 - 5(9)+13}=\sqrt{81-45+13}=\sqrt{49}=\pm7\)

Now, we must find the values of \(\displaystyle \small f(\pm 7)\):

\(\displaystyle \small f(\pm 7)= (\pm 7)^2 +10=49+10=59\)

Because our x term is squared in this function, both values end up being the same. Therefore, 59 is our final answer.

Beginning with the innermost function, we must first solve for \(\displaystyle \small \small g(-9)\):

\(\displaystyle \small g(9)=-\frac{1}{3}(-9)-2=3-2=1\)

We then take this value and plug it into \(\displaystyle \small f(x)\):

\(\displaystyle \small f(1)=\sqrt{(1)^2-4}=\sqrt{-3}\)

This has no value in the real number plane, and the answer is therefore undefined.

Substituting -x into f(x).  This has no effect on the 1st and 3rd terms.  This changes the sign of the middle term.

A polynomial in standard form is written in descending order of the power. The highest power should be first, and the lowest power should be last.

\(\displaystyle \small \small x^4+x^2+6x-2\)

The answer has the powers decreasing from four, to two, to one, to zero.

Expression 5 has the term \(\displaystyle -3\sqrt{x}\), which violates the definition of a polynomial.  The exponent must be a whole number.