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Sagot :
To determine which of the given expressions are polynomials, we need to recall the definition of a polynomial. A polynomial is an algebraic expression consisting of variables and coefficients, with the variables having non-negative integer exponents. The general form of a polynomial in one variable [tex]\( x \)[/tex] is:
[tex]\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \][/tex]
where [tex]\( a_n, a_{n-1}, \ldots, a_1, a_0 \)[/tex] are constants and [tex]\( n \)[/tex] is a non-negative integer.
Let's examine each expression one by one:
### Expression A: [tex]\(\frac{2}{x^3} + x + \frac{1}{2}\)[/tex]
Here, [tex]\(\frac{2}{x^3}\)[/tex] is not a polynomial term because the variable [tex]\( x \)[/tex] is in the denominator, which implies a negative exponent [tex]\( x^{-3} \)[/tex].
### Expression B: [tex]\(\frac{2}{3} x^2 + x + 1\)[/tex]
This expression is a polynomial. All terms have non-negative integer exponents:
- [tex]\(\frac{2}{3} x^2\)[/tex] (exponent 2)
- [tex]\( x \)[/tex] (exponent 1)
- [tex]\( 1 \)[/tex] (constant term, which is [tex]\( x^0 \)[/tex])
### Expression C: [tex]\(x^2 + x + \frac{1}{x^2 + 1}\)[/tex]
The term [tex]\(\frac{1}{x^2 + 1}\)[/tex] is not a polynomial term because it involves a rational function with [tex]\( x \)[/tex] in the denominator.
### Expression D: [tex]\(x^3 + 2x + \sqrt{2}\)[/tex]
This expression is a polynomial. All terms have non-negative integer exponents:
- [tex]\( x^3 \)[/tex] (exponent 3)
- [tex]\( 2x \)[/tex] (exponent 1)
- [tex]\(\sqrt{2}\)[/tex] (constant term, which is [tex]\( x^0 \)[/tex])
### Expression E: [tex]\(x^{\frac{2}{3}} + 0 x + 1\)[/tex]
This expression contains the term [tex]\( x^{\frac{2}{3}} \)[/tex], which has a fractional exponent, thus it is not a polynomial.
Based on the examination, the expressions which are polynomials are:
- Expression B: [tex]\(\frac{2}{3} x^2 + x + 1\)[/tex]
- Expression D: [tex]\(x^3 + 2x + \sqrt{2}\)[/tex]
So, the final result is:
- Expression A: Not a polynomial (False)
- Expression B: Polynomial (True)
- Expression C: Not a polynomial (False)
- Expression D: Polynomial (True)
- Expression E: Not a polynomial (False)
Thus, expressions B and D are polynomials.
[tex]\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \][/tex]
where [tex]\( a_n, a_{n-1}, \ldots, a_1, a_0 \)[/tex] are constants and [tex]\( n \)[/tex] is a non-negative integer.
Let's examine each expression one by one:
### Expression A: [tex]\(\frac{2}{x^3} + x + \frac{1}{2}\)[/tex]
Here, [tex]\(\frac{2}{x^3}\)[/tex] is not a polynomial term because the variable [tex]\( x \)[/tex] is in the denominator, which implies a negative exponent [tex]\( x^{-3} \)[/tex].
### Expression B: [tex]\(\frac{2}{3} x^2 + x + 1\)[/tex]
This expression is a polynomial. All terms have non-negative integer exponents:
- [tex]\(\frac{2}{3} x^2\)[/tex] (exponent 2)
- [tex]\( x \)[/tex] (exponent 1)
- [tex]\( 1 \)[/tex] (constant term, which is [tex]\( x^0 \)[/tex])
### Expression C: [tex]\(x^2 + x + \frac{1}{x^2 + 1}\)[/tex]
The term [tex]\(\frac{1}{x^2 + 1}\)[/tex] is not a polynomial term because it involves a rational function with [tex]\( x \)[/tex] in the denominator.
### Expression D: [tex]\(x^3 + 2x + \sqrt{2}\)[/tex]
This expression is a polynomial. All terms have non-negative integer exponents:
- [tex]\( x^3 \)[/tex] (exponent 3)
- [tex]\( 2x \)[/tex] (exponent 1)
- [tex]\(\sqrt{2}\)[/tex] (constant term, which is [tex]\( x^0 \)[/tex])
### Expression E: [tex]\(x^{\frac{2}{3}} + 0 x + 1\)[/tex]
This expression contains the term [tex]\( x^{\frac{2}{3}} \)[/tex], which has a fractional exponent, thus it is not a polynomial.
Based on the examination, the expressions which are polynomials are:
- Expression B: [tex]\(\frac{2}{3} x^2 + x + 1\)[/tex]
- Expression D: [tex]\(x^3 + 2x + \sqrt{2}\)[/tex]
So, the final result is:
- Expression A: Not a polynomial (False)
- Expression B: Polynomial (True)
- Expression C: Not a polynomial (False)
- Expression D: Polynomial (True)
- Expression E: Not a polynomial (False)
Thus, expressions B and D are polynomials.
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