Let's analyze and re-arrange the terms of the given polynomial for clarity.
We are given the polynomial:
[tex]\[ 10 + 4x^3 - 8x + 7x^2 + 10x^4 \][/tex]
To write this polynomial in standard form, we need to order the terms by descending powers of [tex]\( x \)[/tex]. Here are the steps:
1. Identify the degree of each term: This refers to the exponent of [tex]\( x \)[/tex] in each term:
- [tex]\( 10 \)[/tex]: constant term ([tex]\( x^0 \)[/tex])
- [tex]\( 4x^3 \)[/tex]: [tex]\( x \)[/tex] to the power of 3
- [tex]\( -8x \)[/tex]: [tex]\( x \)[/tex] to the power of 1
- [tex]\( 7x^2 \)[/tex]: [tex]\( x \)[/tex] to the power of 2
- [tex]\( 10x^4 \)[/tex]: [tex]\( x \)[/tex] to the power of 4
2. Arrange the terms in descending order of the exponents:
- The highest exponent is 4, thus the term [tex]\( 10x^4 \)[/tex] comes first.
- Next is [tex]\( x^3 \)[/tex] with the term [tex]\( 4x^3 \)[/tex].
- Then [tex]\( x^2 \)[/tex] with the term [tex]\( 7x^2 \)[/tex].
- Then [tex]\( x \)[/tex] with the term [tex]\( -8x \)[/tex].
- Finally, the constant term [tex]\( 10 \)[/tex].
3. Write the polynomial in standard form:
[tex]\[
10x^4 + 4x^3 + 7x^2 - 8x + 10
\][/tex]
So, the polynomial [tex]\( 10 + 4x^3 - 8x + 7x^2 + 10x^4 \)[/tex] written in standard form is:
[tex]\[
10 + 10x^4 + 4x^3 + 7x^2 - 8x
\][/tex]
