Answer :
To determine which expression represents a linear expression, we need to look for an expression where the variable [tex]\( x \)[/tex] is raised to the power of 1. A linear expression has the form [tex]\( ax + b \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and the highest exponent of the variable [tex]\( x \)[/tex] is 1.
Let's evaluate each option:
1. [tex]\(-17x^4 - 18x^3 + 19x^2 - 20x + 21\)[/tex]:
- This expression includes terms with [tex]\( x^4 \)[/tex], [tex]\( x^3 \)[/tex], [tex]\( x^2 \)[/tex], and [tex]\( x \)[/tex]. The highest exponent is 4, which means this is a polynomial of degree 4, not linear.
2. [tex]\(18x^3 + 19x^2 - 20x + 21\)[/tex]:
- This expression includes terms with [tex]\( x^3 \)[/tex], [tex]\( x^2 \)[/tex], and [tex]\( x \)[/tex]. The highest exponent is 3, making it a cubic polynomial, not linear.
3. [tex]\(23x^2 + 24x - 25\)[/tex]:
- This expression includes terms with [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex]. The highest exponent is 2, making it a quadratic expression, not linear.
4. [tex]\(4x + 4\)[/tex]:
- This expression has the variable [tex]\( x \)[/tex] raised to the power of 1. There are no higher powers of [tex]\( x \)[/tex]. Therefore, this is indeed a linear expression.
Based on the definitions and our analysis, the expression [tex]\(4x + 4\)[/tex] represents a linear expression.
Let's evaluate each option:
1. [tex]\(-17x^4 - 18x^3 + 19x^2 - 20x + 21\)[/tex]:
- This expression includes terms with [tex]\( x^4 \)[/tex], [tex]\( x^3 \)[/tex], [tex]\( x^2 \)[/tex], and [tex]\( x \)[/tex]. The highest exponent is 4, which means this is a polynomial of degree 4, not linear.
2. [tex]\(18x^3 + 19x^2 - 20x + 21\)[/tex]:
- This expression includes terms with [tex]\( x^3 \)[/tex], [tex]\( x^2 \)[/tex], and [tex]\( x \)[/tex]. The highest exponent is 3, making it a cubic polynomial, not linear.
3. [tex]\(23x^2 + 24x - 25\)[/tex]:
- This expression includes terms with [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex]. The highest exponent is 2, making it a quadratic expression, not linear.
4. [tex]\(4x + 4\)[/tex]:
- This expression has the variable [tex]\( x \)[/tex] raised to the power of 1. There are no higher powers of [tex]\( x \)[/tex]. Therefore, this is indeed a linear expression.
Based on the definitions and our analysis, the expression [tex]\(4x + 4\)[/tex] represents a linear expression.
