Certainly! Let's delve into a detailed, step-by-step breakdown of the expression [tex]\( 16x^4 - 4x^2 + 4x - 1 \)[/tex].
### Step 1: Identify the Polynomial
Firstly, we recognize that we have a polynomial expression:
[tex]\[ 16x^4 - 4x^2 + 4x - 1 \][/tex]
### Step 2: Understand Each Term
The expression is composed of several terms:
1. [tex]\( 16x^4 \)[/tex] is a term of degree 4.
2. [tex]\( -4x^2 \)[/tex] is a term of degree 2.
3. [tex]\( 4x \)[/tex] is a term of degree 1.
4. [tex]\( -1 \)[/tex] is a constant term.
### Step 3: Structure of the Polynomial
Given the polynomial [tex]\( 16x^4 - 4x^2 + 4x - 1 \)[/tex], note the following:
- The highest degree term is [tex]\( 16x^4 \)[/tex], where the coefficient is 16 and the degree of the term is 4.
- The next term is [tex]\( -4x^2 \)[/tex] which decreases the degree by 2.
- [tex]\( 4x \)[/tex] follows, decreasing the degree by 1.
- The constant term [tex]\( -1 \)[/tex] completes the polynomial.
### Step 4: Writing the Polynomial
To present the polynomial in its standard form, we ensure that we list the terms in decreasing order of their degrees:
[tex]\[ 16x^4 - 4x^2 + 4x - 1 \][/tex]
Here the terms are already arranged in the correct order, so no reordering is needed.
### Step 5: Recognizing the Coefficients
Each term's coefficient provides essential information:
- For [tex]\( 16x^4 \)[/tex], the coefficient is 16.
- For [tex]\( -4x^2 \)[/tex], the coefficient is -4.
- For [tex]\( 4x \)[/tex], the coefficient is 4.
- The constant term is -1.
### Conclusion:
Thus, the given polynomial, written and analyzed as a mathematical expression, is:
[tex]\[ 16x^4 - 4x^2 + 4x - 1 \][/tex]
This completes our step-by-step breakdown of the polynomial expression.
