To write the polynomial [tex]\( 2 x^2 + x^3 - 3 + 4 x^5 \)[/tex] in standard form, follow these steps:
1. Identify the terms: The given polynomial consists of the following terms:
- [tex]\( 2x^2 \)[/tex]
- [tex]\( x^3 \)[/tex]
- [tex]\(-3\)[/tex]
- [tex]\( 4x^5 \)[/tex]
2. Rewrite the polynomial: Write down the terms explicitly:
- [tex]\( 2x^2 \)[/tex] is a term with [tex]\( x \)[/tex] raised to the power of 2.
- [tex]\( x^3 \)[/tex] is a term with [tex]\( x \)[/tex] raised to the power of 3.
- [tex]\(-3\)[/tex] is a constant term, with no [tex]\( x \)[/tex].
- [tex]\( 4x^5 \)[/tex] is a term with [tex]\( x \)[/tex] raised to the power of 5.
3. Arrange the terms in descending order of their exponents: To place the polynomial in standard form, list the terms from the highest exponent to the lowest:
- The highest exponent term here is [tex]\( 4x^5 \)[/tex], which comes first.
- The next highest exponent is [tex]\( x^3 \)[/tex], which comes second.
- Followed by [tex]\( 2x^2 \)[/tex], which comes third.
- Finally, the constant term [tex]\(-3\)[/tex] comes last.
4. Write the polynomial in standard form:
Combining the terms in descending order of their exponents, we get:
[tex]\[
4x^5 + x^3 + 2x^2 - 3
\][/tex]
Therefore, the polynomial [tex]\( 2 x^2 + x^3 - 3 + 4 x^5 \)[/tex] in standard form is:
[tex]\[
4x^5 + x^3 + 2x^2 - 3
\][/tex]
