Let's analyze each of the given expressions to determine which one represents a quadratic expression. A quadratic expression is a polynomial of degree 2, which means the highest power of the variable [tex]\( x \)[/tex] in the expression is [tex]\( x^2 \)[/tex].
Here are the given expressions:
1. [tex]\( 6x^4 - 5x^3 + 3x^2 - 7x - 8 \)[/tex]
2. [tex]\( 5x^3 + 3x^2 - 7x - 8 \)[/tex]
3. [tex]\( 2x^2 + 3x - 1 \)[/tex]
4. [tex]\( 3x - 1 \)[/tex]
Let's examine the degree of each polynomial:
1. [tex]\( 6x^4 - 5x^3 + 3x^2 - 7x - 8 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^4 \)[/tex].
- Therefore, this is a polynomial of degree 4, not quadratic.
2. [tex]\( 5x^3 + 3x^2 - 7x - 8 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^3 \)[/tex].
- Therefore, this is a polynomial of degree 3, not quadratic.
3. [tex]\( 2x^2 + 3x - 1 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex].
- Therefore, this is a polynomial of degree 2, which makes it a quadratic expression.
4. [tex]\( 3x - 1 \)[/tex]
- The highest power of [tex]\( x \)[/tex] is [tex]\( x \)[/tex].
- Therefore, this is a polynomial of degree 1, not quadratic.
Based on the degrees of the polynomials, the expression that represents a quadratic polynomial is:
[tex]\[ 2x^2 + 3x - 1 \][/tex]
Thus, the correct answer is the third expression.
