Answer :
To determine which of the given polynomials is in standard form, we need to arrange the terms of each polynomial based on the powers of [tex]\( x \)[/tex], from the highest degree to the lowest degree. Let's look at each option:
1. [tex]\(-10x^3 + 25x^2\)[/tex]:
This polynomial is not in standard form because it needs to follow the order of decreasing powers of [tex]\( x \)[/tex]. However, it's already in order since the highest power [tex]\( x^3 \)[/tex] comes first, then [tex]\( x^2 \)[/tex].
2. [tex]\(25x^2 + x^4 - 10x^3\)[/tex]:
In this polynomial, the terms are not in order of decreasing powers of [tex]\( x \)[/tex]. The term [tex]\( x^4 \)[/tex] should come first, followed by [tex]\(-10x^3\)[/tex], then [tex]\( 25x^2 \)[/tex].
3. [tex]\(-10x^3 + 25x^2 + x^4\)[/tex]:
This polynomial also is not in the correct order because the term [tex]\( x^4 \)[/tex] should be first.
4. [tex]\(x^4 + 25x^2 - 10x^3\)[/tex]:
This polynomial is in standard form because the terms are arranged from the largest power [tex]\( x^4 \)[/tex], followed by [tex]\(-10x^3\)[/tex], and then [tex]\( 25x^2 \)[/tex]. However, the terms are not correctly ordered by the coefficients, so the order should be arranged as:
[tex]\[
x^4 - 10x^3 + 25x^2
\][/tex]
Based on the criteria for standard form, option 4 ([tex]\(x^4 - 10x^3 + 25x^2\)[/tex]) can be rearranged correctly, making it display the terms in decreasing powers of [tex]\( x \)[/tex], and thus it is the polynomial in standard form.
1. [tex]\(-10x^3 + 25x^2\)[/tex]:
This polynomial is not in standard form because it needs to follow the order of decreasing powers of [tex]\( x \)[/tex]. However, it's already in order since the highest power [tex]\( x^3 \)[/tex] comes first, then [tex]\( x^2 \)[/tex].
2. [tex]\(25x^2 + x^4 - 10x^3\)[/tex]:
In this polynomial, the terms are not in order of decreasing powers of [tex]\( x \)[/tex]. The term [tex]\( x^4 \)[/tex] should come first, followed by [tex]\(-10x^3\)[/tex], then [tex]\( 25x^2 \)[/tex].
3. [tex]\(-10x^3 + 25x^2 + x^4\)[/tex]:
This polynomial also is not in the correct order because the term [tex]\( x^4 \)[/tex] should be first.
4. [tex]\(x^4 + 25x^2 - 10x^3\)[/tex]:
This polynomial is in standard form because the terms are arranged from the largest power [tex]\( x^4 \)[/tex], followed by [tex]\(-10x^3\)[/tex], and then [tex]\( 25x^2 \)[/tex]. However, the terms are not correctly ordered by the coefficients, so the order should be arranged as:
[tex]\[
x^4 - 10x^3 + 25x^2
\][/tex]
Based on the criteria for standard form, option 4 ([tex]\(x^4 - 10x^3 + 25x^2\)[/tex]) can be rearranged correctly, making it display the terms in decreasing powers of [tex]\( x \)[/tex], and thus it is the polynomial in standard form.
