Question

Five students makes claims and share them about two functions f(x) = x² - 2x - 1

and g(x) = (x + 1)²:
Shawn: "With g(x) = (x + 1)², we could rewrite that as ² +2x+1."
Kolby: "If we add them g (x) + f(x), we get 2x².
Shannon: "If we subtract them f (x) - g(x). the setup would be
x² − 2x − 1 − (x² + 1).
Jolie: "If we add them f(x) + g(x), we get 2x² - 2x."
Antoine: "The function g(x) = (x + 1)² can be rewritten as x² + 1² = x² + 1
since the exponent is distributed."
Identify each student's claim as TRUE or FALSE.

Answer 1

Here are the evaluations of each student's claim:

Shawn: FALSE

Shawn's claim is FALSE. He made a mistake in his statement. The correct expansion of g(x) = (x + 1)² is x² + 2x + 1, not ² + 2x + 1.

Kolby: TRUE

Kolby's claim is TRUE. When you add g(x) and f(x), you indeed get 2x²:

g(x) + f(x) = (x + 1)² + (x² - 2x - 1) = x² + 2x + 1 + x² - 2x - 1 = 2x².

Shannon: TRUE

Shannon's claim is TRUE. When you subtract g(x) from f(x), you indeed get x² - 2x - 1 - (x² + 1) = x² - 2x - 1 - x² - 1 = -2.

Jolie: FALSE

Jolie's claim is FALSE. Adding f(x) and g(x) does not result in 2x² - 2x. It results in 2x².

Antoine: FALSE

Antoine's claim is FALSE. The exponent 1 in g(x) = (x + 1)² does not get distributed to each term. The correct expansion of g(x) is x² + 2x + 1, not x² + 1.

In summary:

Kolby and Shannon made TRUE claims.

Shawn, Jolie, and Antoine made FALSE claims.