QQuestionMathematics
QuestionMathematics
Which of the following are polynomials?
\begin{array}{ll}
\square & \text { A. } \frac{2}{3} x^{2}+x+ 1 \\
\square & \text { B. } \frac{2}{x^{3}}+x+\frac{1}{2} \\
\square & \text { C. } x^{2}+ 2 x+\sqrt{2} \\
\square & \text { D. } \frac{2}{x^{3}}+ 0 x+ 1 \\
\square & \text { E. } x^{2}+x+\frac{1}{x^{2}+ 1}
\end{array}
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Answer
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Step 1:Let's solve this step by step by checking each expression against the definition of a polynomial.
Step 2:: Definition of a Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients, where: - Only whole number (non-negative integer) exponents are allowed - No variables in the denominator - No radicals or fractional exponents of variables
Step 3:: Analyze Option A: \frac{2}{3}x^{2}+x+ 1
- All terms have non-negative integer exponents - Coefficients are allowed to be fractions - ✓ This IS a polynomial
Step 4:: Analyze Option B: \frac{2}{x^{3}}+x+\frac{1}{2}
- Contains $$\frac{2}{x^{3}}$$, which has a variable in the denominator
- ✗ NOT a polynomial
Step 5:: Analyze Option C: x^{2}+ 2x+\sqrt{2}
- Constant term $$\sqrt{2}$$ is allowed
- All terms have non-negative integer exponents - ✓ This IS a polynomial
Step 6:: Analyze Option D: \frac{2}{x^{3}}+ 0x+ 1
- Contains $$\frac{2}{x^{3}}$$, which has a variable in the denominator
- ✗ NOT a polynomial
Step 7:: Analyze Option E: x^{2}+x+\frac{1}{x^{2}+ 1}
- Contains $$\frac{1}{x^{2}+1}$$, which has a variable in the denominator
- ✗ NOT a polynomial
Final Answer
The polynomials are A and C.
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