Sure, let's break down and analyze the polynomial expression step by step:
[tex]\[ 16a^4 - 4a^2 - 4a - 1 \][/tex]
1. Identify the Degree of the Polynomial:
The polynomial [tex]\( 16a^4 - 4a^2 - 4a - 1 \)[/tex] is given. The highest power of [tex]\( a \)[/tex] is [tex]\( 4 \)[/tex]. Therefore, the degree of this polynomial is [tex]\( 4 \)[/tex].
2. Identify the Coefficients:
The coefficients of the terms in the polynomial are:
- For [tex]\( a^4 \)[/tex], the coefficient is [tex]\( 16 \)[/tex].
- For [tex]\( a^2 \)[/tex], the coefficient is [tex]\( -4 \)[/tex].
- For [tex]\( a \)[/tex], the coefficient is [tex]\( -4 \)[/tex].
- The constant term is [tex]\( -1 \)[/tex].
3. Identify the Terms:
The polynomial can be broken down into individual terms as:
- [tex]\( 16a^4 \)[/tex] is the leading term with the highest exponent.
- [tex]\( -4a^2 \)[/tex] is the quadratic term.
- [tex]\( -4a \)[/tex] is the linear term.
- [tex]\( -1 \)[/tex] is the constant term.
4. Standard Form:
The polynomial is already in its standard form:
[tex]\[ 16a^4 - 4a^2 - 4a - 1 \][/tex]
This means the terms are arranged in descending order of the exponent of [tex]\( a \)[/tex].
5. Possible Simplifications:
It's good practice to check if the expression can be factored or simplified further. In this case, our polynomial:
[tex]\[ 16a^4 - 4a^2 - 4a - 1 \][/tex]
does not have any common factors amongst all terms and does not easily factor into simpler polynomials or binomials by inspection.
So, we recognize the polynomial as:
[tex]\[ 16a^4 - 4a^2 - 4a - 1 \][/tex]
This is a quartic polynomial in its standard form with the coefficients and terms identified and analyzed.
