Question

martha wrote an example of a quadratic function for a homework assignment. the function she wrote is shown.
$f(x) = 5x^3 + 2x^2 + 7x - 3$
what possible changes can martha make to correct her homework assignment? select two options.

• the first term, $5x^3$, can be eliminated.
• the exponent on the first term, $5x^3$, can be changed to a 2 and then combined with the second term, $2x^2$.
• the exponent on the second term, $2x^2$, can be changed to a 3 and then combined with the first term, $5x^3$.
• the constant, $-3$, can be changed to a variable.
• the $7x$ can be eliminated.

Explanation:

Step1: Recall Quadratic Function Definition

A quadratic function has the form \( f(x) = ax^2 + bx + c \), where \( a
eq 0 \) and the highest degree (exponent) is 2. Martha's function is \( f(x) = 5x^3 + 2x^2 + 7x - 3 \), which is a cubic function (degree 3) currently.

Step2: Analyze Each Option

• Option 1: Eliminating \( 5x^3 \) (the cubic term) leaves \( f(x) = 2x^2 + 7x - 3 \), which is quadratic. Valid.
• Option 2: Changing the exponent of \( 5x^3 \) to 2 gives \( 5x^2 \), which can combine with \( 2x^2 \) (like terms) to form \( 7x^2 + 7x - 3 \), a quadratic. Valid.
• Option 3: Changing the exponent of \( 2x^2 \) to 3 gives \( 2x^3 \), combining with \( 5x^3 \) makes it a cubic function (still degree 3). Invalid.
• Option 4: Changing the constant \(-3\) to a variable doesn't affect the degree (still has \( x^3 \)). Invalid.
• Option 5: Eliminating \( 7x \) leaves \( f(x) = 5x^3 + 2x^2 - 3 \), still cubic. Invalid.

Answer:

The first term, \( 5x^3 \), can be eliminated.
The exponent on the first term, \( 5x^3 \), can be changed to a 2 and then combined with the second term, \( 2x^2 \).