Answer :
To determine which expression represents a linear expression, we need to understand what a linear expression is. A linear expression in one variable, like [tex]\(x\)[/tex], is an algebraic expression of the form [tex]\(ax + b\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are constants, and [tex]\(x\)[/tex] is the variable. This means the highest power of the variable [tex]\(x\)[/tex] that a linear expression can have is 1.
Let's analyze each given expression:
1. [tex]\(-17x^4 - 18x^3 + 19x^2 - 20x + 21\)[/tex]:
- This expression has terms with [tex]\(x\)[/tex] raised to powers higher than 1 (like [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex]), so it is not linear.
2. [tex]\(18x^3 + 19x^2 - 20x + 21\)[/tex]:
- This expression also includes terms where [tex]\(x\)[/tex] is raised to powers higher than 1 (such as [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex]), so it is not linear.
3. [tex]\(23x^2 + 24x - 25\)[/tex]:
- This expression includes a term with [tex]\(x\)[/tex] squared ([tex]\(x^2\)[/tex]), so it is not linear.
4. [tex]\(4x + 4\)[/tex]:
- This expression has [tex]\(x\)[/tex] raised to the power of 1, and it fits the form [tex]\(ax + b\)[/tex], where [tex]\(a = 4\)[/tex] and [tex]\(b = 4\)[/tex]. Thus, it is a linear expression.
Therefore, the expression that represents a linear expression is [tex]\(4x + 4\)[/tex].
Let's analyze each given expression:
1. [tex]\(-17x^4 - 18x^3 + 19x^2 - 20x + 21\)[/tex]:
- This expression has terms with [tex]\(x\)[/tex] raised to powers higher than 1 (like [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex]), so it is not linear.
2. [tex]\(18x^3 + 19x^2 - 20x + 21\)[/tex]:
- This expression also includes terms where [tex]\(x\)[/tex] is raised to powers higher than 1 (such as [tex]\(x^3\)[/tex] and [tex]\(x^2\)[/tex]), so it is not linear.
3. [tex]\(23x^2 + 24x - 25\)[/tex]:
- This expression includes a term with [tex]\(x\)[/tex] squared ([tex]\(x^2\)[/tex]), so it is not linear.
4. [tex]\(4x + 4\)[/tex]:
- This expression has [tex]\(x\)[/tex] raised to the power of 1, and it fits the form [tex]\(ax + b\)[/tex], where [tex]\(a = 4\)[/tex] and [tex]\(b = 4\)[/tex]. Thus, it is a linear expression.
Therefore, the expression that represents a linear expression is [tex]\(4x + 4\)[/tex].
