To analyze the polynomial [tex]\( f(x) = 2x^5 - 3x^4 + x - 6 \)[/tex], we will identify four key characteristics: the degree, the leading term, the leading coefficient, and the constant term.
1. Degree of the Polynomial:
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] in the polynomial. In this case, the highest power of [tex]\( x \)[/tex] is [tex]\( 5 \)[/tex] (from the term [tex]\( 2x^5 \)[/tex]). Therefore, the degree of the polynomial is:
[tex]$ \text{Degree} = 5 $[/tex]
2. Leading Term:
The leading term of a polynomial is the term with the highest power of [tex]\( x \)[/tex]. Here, the term with the highest power is [tex]\( 2x^5 \)[/tex]. Therefore, the leading term is:
[tex]$ \text{Leading Term} = 2x^5 $[/tex]
3. Leading Coefficient:
The leading coefficient is the coefficient of the leading term. For the leading term [tex]\( 2x^5 \)[/tex], the coefficient is [tex]\( 2 \)[/tex]. Therefore, the leading coefficient is:
[tex]$ \text{Leading Coefficient} = 2 $[/tex]
4. Constant Term:
The constant term in a polynomial is the term that does not contain the variable [tex]\( x \)[/tex]. Here, the constant term is [tex]\( -6 \)[/tex]. Therefore, the constant term is:
[tex]$ \text{Constant Term} = -6 $[/tex]
Putting it all together, we have:
- The degree of the given polynomial is [tex]\( 5 \)[/tex].
- The leading term is [tex]\( 2x^5 \)[/tex].
- The leading coefficient is [tex]\( 2 \)[/tex].
- The constant term is [tex]\( -6 \)[/tex].
Substituting these into the blanks, we get:
The degree of the given polynomial is [tex]\( 5 \)[/tex].
The leading term is [tex]\( 2x^5 \)[/tex].
The leading coefficient is [tex]\( 2 \)[/tex].
The constant term is [tex]\( -6 \)[/tex].
