Answer :
Let's analyze the given mathematical expressions and categorize them according to their types:
1. [tex]\(x^{-5}-5 x^{-4}+4 x^{-3}-3 x^{-2}+2 x^{-1}-1 \sqrt[4]{x}-\sqrt[3]{x}+4 \sqrt{x}-8 x+16\)[/tex]
2. [tex]\(x^3-7 x^2+9 x-5 x^4-20\)[/tex]
3. [tex]\(x^5-5 x^4+4 x^3-3 x^2+2 x-1\)[/tex]
4. [tex]\(3 x^2-5 x^4+2 x-12\)[/tex]
5. [tex]\(\frac{4}{x^4}+\frac{3}{x^3}-\frac{2}{x^2}-1\)[/tex]
Here are the categories for the expressions:
1. Polynomial Expressions:
- A polynomial is an algebraic expression consisting of variables and coefficients, combined using only addition, subtraction, and multiplication (without division by variables). All the exponents of the variable in a polynomial are whole numbers.
Expression analysis:
- [tex]\(x^3-7 x^2+9 x-5 x^4-20\)[/tex]:
- This consists of terms [tex]\(x^4, x^3, x^2, x,\)[/tex] and a constant, but not in a simplified form. Let's rewrite it: [tex]\(-5x^4 + x^3 - 7x^2 + 9x - 20\)[/tex]. This is a polynomial.
- [tex]\(x^5-5 x^4+4 x^3-3 x^2+2 x-1\)[/tex]:
- This consists of terms [tex]\(x^5, x^4, x^3, x^2, x,\)[/tex] and a constant. This is a polynomial.
- [tex]\(3 x^2-5 x^4+2 x-12\)[/tex]:
- This consists of terms [tex]\(x^4, x^2, x,\)[/tex] and a constant. This is a polynomial.
Thus, these expressions are polynomials:
- [tex]\(x^3-7 x^2+9 x-5 x^4-20\)[/tex]
- [tex]\(x^5-5 x^4+4 x^3-3 x^2+2 x-1\)[/tex]
- [tex]\(3 x^2-5 x^4+2 x-12\)[/tex]
2. Rational Expressions:
- A rational expression is a ratio or fraction of two polynomials.
Expression analysis:
- [tex]\(\frac{4}{x^4}+\frac{3}{x^3}-\frac{2}{x^2}-1\)[/tex]:
- This consists of terms that are quotients of polynomials. This is a rational expression.
Thus, this expression is a rational expression:
- [tex]\(\frac{4}{x^4}+\frac{3}{x^3}-\frac{2}{x^2}-1\)[/tex]
3. Miscellaneous Expressions:
- Contains terms that involve exponents, roots, or other types of relationships not strictly polynomial or rational.
Expression analysis:
- [tex]\(x^{-5}-5 x^{-4}+4 x^{-3}-3 x^{-2}+2 x^{-1}-1 \sqrt[4]{x}-\sqrt[3]{x}+4 \sqrt{x}-8 x+16\)[/tex]:
- This contains negative exponents, roots, and polynomial terms. This does not fit purely into the category of polynomials or rational expressions.
Thus, this expression is categorized as a miscellaneous expression:
- [tex]\(x^{-5}-5 x^{-4}+4 x^{-3}-3 x^{-2}+2 x^{-1}-1 \sqrt[4]{x}-\sqrt[3]{x}+4 \sqrt{x}-8 x+16\)[/tex]
Thus, the completed table is:
Polynomials:
- [tex]\(x^3-7 x^2+9 x-5 x^4-20\)[/tex]
- [tex]\(x^5-5 x^4+4 x^3-3 x^2+2 x-1\)[/tex]
- [tex]\(3 x^2-5 x^4+2 x-12\)[/tex]
Rational Expressions:
- [tex]\(\frac{4}{x^4}+\frac{3}{x^3}-\frac{2}{x^2}-1\)[/tex]
Miscellaneous Expressions:
- [tex]\(x^{-5}-5 x^{-4}+4 x^{-3}-3 x^{-2}+2 x^{-1}-1 \sqrt[4]{x}-\sqrt[3]{x}+4 \sqrt{x}-8 x+16\)[/tex]
1. [tex]\(x^{-5}-5 x^{-4}+4 x^{-3}-3 x^{-2}+2 x^{-1}-1 \sqrt[4]{x}-\sqrt[3]{x}+4 \sqrt{x}-8 x+16\)[/tex]
2. [tex]\(x^3-7 x^2+9 x-5 x^4-20\)[/tex]
3. [tex]\(x^5-5 x^4+4 x^3-3 x^2+2 x-1\)[/tex]
4. [tex]\(3 x^2-5 x^4+2 x-12\)[/tex]
5. [tex]\(\frac{4}{x^4}+\frac{3}{x^3}-\frac{2}{x^2}-1\)[/tex]
Here are the categories for the expressions:
1. Polynomial Expressions:
- A polynomial is an algebraic expression consisting of variables and coefficients, combined using only addition, subtraction, and multiplication (without division by variables). All the exponents of the variable in a polynomial are whole numbers.
Expression analysis:
- [tex]\(x^3-7 x^2+9 x-5 x^4-20\)[/tex]:
- This consists of terms [tex]\(x^4, x^3, x^2, x,\)[/tex] and a constant, but not in a simplified form. Let's rewrite it: [tex]\(-5x^4 + x^3 - 7x^2 + 9x - 20\)[/tex]. This is a polynomial.
- [tex]\(x^5-5 x^4+4 x^3-3 x^2+2 x-1\)[/tex]:
- This consists of terms [tex]\(x^5, x^4, x^3, x^2, x,\)[/tex] and a constant. This is a polynomial.
- [tex]\(3 x^2-5 x^4+2 x-12\)[/tex]:
- This consists of terms [tex]\(x^4, x^2, x,\)[/tex] and a constant. This is a polynomial.
Thus, these expressions are polynomials:
- [tex]\(x^3-7 x^2+9 x-5 x^4-20\)[/tex]
- [tex]\(x^5-5 x^4+4 x^3-3 x^2+2 x-1\)[/tex]
- [tex]\(3 x^2-5 x^4+2 x-12\)[/tex]
2. Rational Expressions:
- A rational expression is a ratio or fraction of two polynomials.
Expression analysis:
- [tex]\(\frac{4}{x^4}+\frac{3}{x^3}-\frac{2}{x^2}-1\)[/tex]:
- This consists of terms that are quotients of polynomials. This is a rational expression.
Thus, this expression is a rational expression:
- [tex]\(\frac{4}{x^4}+\frac{3}{x^3}-\frac{2}{x^2}-1\)[/tex]
3. Miscellaneous Expressions:
- Contains terms that involve exponents, roots, or other types of relationships not strictly polynomial or rational.
Expression analysis:
- [tex]\(x^{-5}-5 x^{-4}+4 x^{-3}-3 x^{-2}+2 x^{-1}-1 \sqrt[4]{x}-\sqrt[3]{x}+4 \sqrt{x}-8 x+16\)[/tex]:
- This contains negative exponents, roots, and polynomial terms. This does not fit purely into the category of polynomials or rational expressions.
Thus, this expression is categorized as a miscellaneous expression:
- [tex]\(x^{-5}-5 x^{-4}+4 x^{-3}-3 x^{-2}+2 x^{-1}-1 \sqrt[4]{x}-\sqrt[3]{x}+4 \sqrt{x}-8 x+16\)[/tex]
Thus, the completed table is:
Polynomials:
- [tex]\(x^3-7 x^2+9 x-5 x^4-20\)[/tex]
- [tex]\(x^5-5 x^4+4 x^3-3 x^2+2 x-1\)[/tex]
- [tex]\(3 x^2-5 x^4+2 x-12\)[/tex]
Rational Expressions:
- [tex]\(\frac{4}{x^4}+\frac{3}{x^3}-\frac{2}{x^2}-1\)[/tex]
Miscellaneous Expressions:
- [tex]\(x^{-5}-5 x^{-4}+4 x^{-3}-3 x^{-2}+2 x^{-1}-1 \sqrt[4]{x}-\sqrt[3]{x}+4 \sqrt{x}-8 x+16\)[/tex]
