A proposed linear factor has to the first power, and the Factor Theorem turns it into a zero test. Set the whole factor equal to , then define the polynomial in Desmos and evaluate it at that input. An output of means the factor divides evenly. For , account for the number multiplying when finding the input.
Hints
- Hint 1
A factor divides a polynomial evenly. For a proposed linear factor, the polynomial must output at the input that zeros the factor. Which input makes equal ?
- Hint 2
The zero of is the input that makes the whole expression equal . Solve , including the on , before checking that input in the polynomial.
Step-by-step
Test the zero of each factor
Step 1Find the test input for I
A factor divides a polynomial evenly. The Factor Theorem says a linear factor works exactly when the polynomial is where that factor is . Set I equal to :
Add to both sides:
- Step 2
Test I in Desmos
Define as the given polynomial in Desmos, then type . Desmos prints , so is a factor. That settles I, but II has a different test input.
- Step 3
Find the test input for II
For II, the changes the input. Set the whole factor equal to :
Add to both sides:
Divide by :
Testing instead would not test this factor.
- Step 4
Test II and choose
Keep the earlier Desmos lines and type . Desmos prints , so is also a factor. Test each proposed linear factor at the input that makes that whole factor . Both I and II are factors. Choice C.