Let's analyze the given polynomial [tex]\( 1 - 7x^3 + \frac{x}{9} \)[/tex] systematically to answer the questions:
### Determine the Type of Polynomial
A polynomial is classified based on the highest power (degree) of the variable [tex]\( x \)[/tex].
- In the given polynomial [tex]\( 1 - 7x^3 + \frac{x}{9} \)[/tex], the term with the highest power of [tex]\( x \)[/tex] is [tex]\( -7x^3 \)[/tex], which has a degree of 3.
- Therefore, this is a cubic polynomial because the highest power of [tex]\( x \)[/tex] is 3.
### Count the Number of Terms
The terms of a polynomial are separated by addition or subtraction signs.
- The given polynomial can be broken down into three terms: [tex]\( 1 \)[/tex], [tex]\( -7x^3 \)[/tex], and [tex]\( \frac{x}{9} \)[/tex].
- Therefore, the polynomial has 3 terms.
### Identify the Constant Term
The constant term is the term that does not contain the variable [tex]\( x \)[/tex].
- In [tex]\( 1 - 7x^3 + \frac{x}{9} \)[/tex], the term [tex]\( 1 \)[/tex] does not contain [tex]\( x \)[/tex].
- Therefore, the constant term is [tex]\( 1 \)[/tex].
### Determine the Leading Term
The leading term is the term with the highest power of [tex]\( x \)[/tex].
- In [tex]\( 1 - 7x^3 + \frac{x}{9} \)[/tex], the term with the highest power of [tex]\( x \)[/tex] is [tex]\( -7x^3 \)[/tex].
- Therefore, the leading term is [tex]\( -7x^3 \)[/tex].
### Determine the Leading Coefficient
The leading coefficient is the coefficient of the leading term.
- In [tex]\( -7x^3 \)[/tex], the coefficient is [tex]\( -7 \)[/tex].
- Therefore, the leading coefficient is [tex]\( -7 \)[/tex].
### Putting It All Together
- The expression represents a cubic polynomial with 3 terms.
- The constant term is 1.
- The leading term is -7x^3.
- The leading coefficient is -7.
So, the completed sentence is:
The expression represents a cubic polynomial with 3 terms. The constant term is 1 , the leading term is -7x^3 , and the leading coefficient is -7.
