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Sagot :
To determine which of the given algebraic expressions is a polynomial with a degree of 3, let's analyze each option step-by-step.
### Expression 1: [tex]\( 4x^3 - \frac{2}{x} \)[/tex]
1. [tex]\( 4x^3 \)[/tex] is a term with [tex]\( x \)[/tex] raised to the power of 3, which is the highest power of [tex]\( x \)[/tex] in this term.
2. [tex]\( \frac{2}{x} \)[/tex] is a term with [tex]\( x \)[/tex] in the denominator, which is equivalent to [tex]\( 2x^{-1} \)[/tex].
For an expression to be a polynomial, all the exponents of the variables must be non-negative integers. Since [tex]\( \frac{2}{x} \)[/tex] or [tex]\( 2x^{-1} \)[/tex] contains a negative exponent, this expression is not a polynomial.
### Expression 2: [tex]\( 2y^3 + 5y^2 - 5y \)[/tex]
1. [tex]\( 2y^3 \)[/tex] is a term with [tex]\( y \)[/tex] raised to the power of 3, which is the highest power of [tex]\( y \)[/tex] in this expression.
2. [tex]\( 5y^2 \)[/tex] is a term with [tex]\( y \)[/tex] raised to the power of 2.
3. [tex]\( -5y \)[/tex] is a term with [tex]\( y \)[/tex] raised to the power of 1.
All these terms have non-negative integer exponents, which means this expression is a polynomial. The highest power of [tex]\( y \)[/tex] here is 3, making it a polynomial of degree 3.
### Expression 3: [tex]\( 3y^3 - \sqrt{4y} \)[/tex]
1. [tex]\( 3y^3 \)[/tex] is a term with [tex]\( y \)[/tex] raised to the power of 3.
2. [tex]\( \sqrt{4y} \)[/tex] can be rewritten as [tex]\( (4y)^{\frac{1}{2}} \)[/tex] or [tex]\( 4^{\frac{1}{2}} y^{\frac{1}{2}} \)[/tex]. This involves a fractional exponent (1/2).
Since [tex]\( \sqrt{4y} \)[/tex] includes a fractional exponent, this expression is not a polynomial.
### Expression 4: [tex]\( -xy\sqrt{6} \)[/tex]
1. The term [tex]\( xy \)[/tex] involves the product of two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
2. [tex]\( \sqrt{6} \)[/tex] is a constant, which does not affect the degree but the product term [tex]\( xy \)[/tex], which means the exponents are summed for polynomial terms.
For this expression, the presence of the variables' product [tex]\( xy \)[/tex] and any radical (even if a constant) makes the term not a standard form polynomial since each term in a polynomial should be in the form of [tex]\( ax^n \)[/tex] where [tex]\( n \)[/tex] is a whole number.
### Conclusion:
The expression that is a polynomial with a degree of 3 is:
[tex]\[ 2y^3 + 5y^2 - 5y \][/tex]
Thus, the correct option is the second expression [tex]\(2 y^3 + 5 y^2 - 5 y\)[/tex].
### Expression 1: [tex]\( 4x^3 - \frac{2}{x} \)[/tex]
1. [tex]\( 4x^3 \)[/tex] is a term with [tex]\( x \)[/tex] raised to the power of 3, which is the highest power of [tex]\( x \)[/tex] in this term.
2. [tex]\( \frac{2}{x} \)[/tex] is a term with [tex]\( x \)[/tex] in the denominator, which is equivalent to [tex]\( 2x^{-1} \)[/tex].
For an expression to be a polynomial, all the exponents of the variables must be non-negative integers. Since [tex]\( \frac{2}{x} \)[/tex] or [tex]\( 2x^{-1} \)[/tex] contains a negative exponent, this expression is not a polynomial.
### Expression 2: [tex]\( 2y^3 + 5y^2 - 5y \)[/tex]
1. [tex]\( 2y^3 \)[/tex] is a term with [tex]\( y \)[/tex] raised to the power of 3, which is the highest power of [tex]\( y \)[/tex] in this expression.
2. [tex]\( 5y^2 \)[/tex] is a term with [tex]\( y \)[/tex] raised to the power of 2.
3. [tex]\( -5y \)[/tex] is a term with [tex]\( y \)[/tex] raised to the power of 1.
All these terms have non-negative integer exponents, which means this expression is a polynomial. The highest power of [tex]\( y \)[/tex] here is 3, making it a polynomial of degree 3.
### Expression 3: [tex]\( 3y^3 - \sqrt{4y} \)[/tex]
1. [tex]\( 3y^3 \)[/tex] is a term with [tex]\( y \)[/tex] raised to the power of 3.
2. [tex]\( \sqrt{4y} \)[/tex] can be rewritten as [tex]\( (4y)^{\frac{1}{2}} \)[/tex] or [tex]\( 4^{\frac{1}{2}} y^{\frac{1}{2}} \)[/tex]. This involves a fractional exponent (1/2).
Since [tex]\( \sqrt{4y} \)[/tex] includes a fractional exponent, this expression is not a polynomial.
### Expression 4: [tex]\( -xy\sqrt{6} \)[/tex]
1. The term [tex]\( xy \)[/tex] involves the product of two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
2. [tex]\( \sqrt{6} \)[/tex] is a constant, which does not affect the degree but the product term [tex]\( xy \)[/tex], which means the exponents are summed for polynomial terms.
For this expression, the presence of the variables' product [tex]\( xy \)[/tex] and any radical (even if a constant) makes the term not a standard form polynomial since each term in a polynomial should be in the form of [tex]\( ax^n \)[/tex] where [tex]\( n \)[/tex] is a whole number.
### Conclusion:
The expression that is a polynomial with a degree of 3 is:
[tex]\[ 2y^3 + 5y^2 - 5y \][/tex]
Thus, the correct option is the second expression [tex]\(2 y^3 + 5 y^2 - 5 y\)[/tex].
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