Answer :
Certainly! Let’s analyze the expression [tex]\( 5 x^3 - 6 x^2 - \frac{25}{y} + 18 \)[/tex] and determine which statements are true step-by-step.
1. The expression is a polynomial in [tex]\( x \)[/tex].
- A polynomial in [tex]\( x \)[/tex] is an expression that consists of terms of the form [tex]\( a_n x^n \)[/tex], where [tex]\( a_n \)[/tex] are constant coefficients and [tex]\( n \)[/tex] are non-negative integers.
- In the given expression, terms [tex]\( 5 x^3 \)[/tex] and [tex]\( -6 x^2 \)[/tex] fit this definition.
- However, the term [tex]\( -\frac{25}{y} \)[/tex] is not a polynomial term in [tex]\( x \)[/tex] because it introduces a variable [tex]\( y \)[/tex] in the denominator.
- Therefore, this expression is not purely a polynomial in [tex]\( x \)[/tex].
2. The expression is a rational function in [tex]\( x \)[/tex].
- A rational function in [tex]\( x \)[/tex] is a ratio of two polynomial functions in [tex]\( x \)[/tex].
- Since our expression includes [tex]\( -\frac{25}{y} \)[/tex], it introduces a dependence on the variable [tex]\( y \)[/tex], which is not a polynomial term involving [tex]\( x \)[/tex] alone.
- Hence, our expression is not a rational function in [tex]\( x \)[/tex].
3. The expression has a constant term.
- A constant term in an expression is a term that does not change with the variables [tex]\( x \)[/tex] or [tex]\( y \)[/tex].
- In our given expression, [tex]\( +18 \)[/tex] is a constant term because it does not depend on [tex]\( x \)[/tex] or [tex]\( y \)[/tex].
- Therefore, this statement is true.
4. The expression is a rational function in [tex]\( y \)[/tex].
- A rational function in [tex]\( y \)[/tex] is a ratio of two polynomial functions in [tex]\( y \)[/tex].
- Considering [tex]\( y \)[/tex] as the variable, the term [tex]\( -\frac{25}{y} \)[/tex] makes the expression a rational function in [tex]\( y \)[/tex], since the denominator contains the variable [tex]\( y \)[/tex].
- Thus, this statement is true.
To summarize, the true statements are:
- The expression has a constant term.
- The expression is a rational function in [tex]\( y \)[/tex].
Therefore, the correct statements are Statements 3 and 4.
1. The expression is a polynomial in [tex]\( x \)[/tex].
- A polynomial in [tex]\( x \)[/tex] is an expression that consists of terms of the form [tex]\( a_n x^n \)[/tex], where [tex]\( a_n \)[/tex] are constant coefficients and [tex]\( n \)[/tex] are non-negative integers.
- In the given expression, terms [tex]\( 5 x^3 \)[/tex] and [tex]\( -6 x^2 \)[/tex] fit this definition.
- However, the term [tex]\( -\frac{25}{y} \)[/tex] is not a polynomial term in [tex]\( x \)[/tex] because it introduces a variable [tex]\( y \)[/tex] in the denominator.
- Therefore, this expression is not purely a polynomial in [tex]\( x \)[/tex].
2. The expression is a rational function in [tex]\( x \)[/tex].
- A rational function in [tex]\( x \)[/tex] is a ratio of two polynomial functions in [tex]\( x \)[/tex].
- Since our expression includes [tex]\( -\frac{25}{y} \)[/tex], it introduces a dependence on the variable [tex]\( y \)[/tex], which is not a polynomial term involving [tex]\( x \)[/tex] alone.
- Hence, our expression is not a rational function in [tex]\( x \)[/tex].
3. The expression has a constant term.
- A constant term in an expression is a term that does not change with the variables [tex]\( x \)[/tex] or [tex]\( y \)[/tex].
- In our given expression, [tex]\( +18 \)[/tex] is a constant term because it does not depend on [tex]\( x \)[/tex] or [tex]\( y \)[/tex].
- Therefore, this statement is true.
4. The expression is a rational function in [tex]\( y \)[/tex].
- A rational function in [tex]\( y \)[/tex] is a ratio of two polynomial functions in [tex]\( y \)[/tex].
- Considering [tex]\( y \)[/tex] as the variable, the term [tex]\( -\frac{25}{y} \)[/tex] makes the expression a rational function in [tex]\( y \)[/tex], since the denominator contains the variable [tex]\( y \)[/tex].
- Thus, this statement is true.
To summarize, the true statements are:
- The expression has a constant term.
- The expression is a rational function in [tex]\( y \)[/tex].
Therefore, the correct statements are Statements 3 and 4.
