Let’s consider the given polynomial:
[tex]\[
6x + 5x^4 - 10x + 8x^3 - 7x^2 + 9^2
\][/tex]
To write this polynomial in standard form, we need to arrange the terms by decreasing powers of [tex]\( x \)[/tex].
1. First, identify and group the like terms:
- The [tex]\( x^4 \)[/tex] term: [tex]\( 5x^4 \)[/tex]
- The [tex]\( x^3 \)[/tex] term: [tex]\( 8x^3 \)[/tex]
- The [tex]\( x^2 \)[/tex] term: [tex]\( -7x^2 \)[/tex]
- The [tex]\( x \)[/tex] terms: [tex]\( 6x \)[/tex] and [tex]\( -10x \)[/tex]
- The constant term: [tex]\( 9^2 \)[/tex]
2. Rearrange the terms in order of decreasing powers of [tex]\( x \)[/tex]:
[tex]\[
5x^4 + 8x^3 - 7x^2 + 6x - 10x + 9^2
\][/tex]
3. Combine the like terms:
[tex]\[
5x^4 + 8x^3 - 7x^2 + (-4x) + 9^2
\][/tex]
4. Simplify the constant term:
[tex]\[
9^2 = 81
\][/tex]
So, the polynomial in standard form is:
[tex]\[
5x^4 + 8x^3 - 7x^2 - 4x + 81
\][/tex]
Now, let’s identify the specific details asked:
- The constant term: The constant term is the term with no [tex]\( x \)[/tex] attached to it, which is [tex]\( 81 \)[/tex].
- The coefficient of the linear term: The linear term is the term with [tex]\( x \)[/tex] to the first power, which in this case is [tex]\( -4x \)[/tex]. Thus, the coefficient of the linear term is [tex]\( -4 \)[/tex].
Therefore:
- The constant term is [tex]\( 81 \)[/tex].
- The coefficient of the linear term is [tex]\( -4 \)[/tex].
