Rational Descartes Factor and Remainder Theorems

Cam divided (x⁴ + 3x² - 4x - 2) by factor of (x-2) using synthetic division. His work is shown above. Which best describes his mistake?

• Cam wrote the remainder incorrectly.

• Cam did not use a zero place holder for the x³ term.

• Cam added instead of subtracting the rows.

• Cam should have used -2 as his division since the factor was x-2.

Is (x-2) a factor of             f(x)= x³-8x²+14x-4?

• Yes, (x-2) is a factor. There is a remainder.

• No, (x-2) is  not a factor. The remainder is zero.

• Yes, (x-2) is a factor. The remainder is zero.

• No, (x-2) is  not a factor. There is a remainder. 

The table gives selected values of a polynomial p(x).
Based on the table and the Remainder Theorem, which of the following must be a factor of the polynomial p(x)?

Is (x-4) a factor of (x³ +x² -16x-16)?

• YES

• NO

Is (x-3) a factor of
(x³ - 3x² + 2x + 2)?

• Yes

• No

• A
• B
• C
• D

What is the remainder when f(x) is divided by (x - 3)? ​ (a)  

Is (x - 3) a factor of f(x)? ​ (b)  

• r=0

• Yes, the remainder is 0

• r=-72

• r=-153

• r=48

• r=-6

• r=10

• r=-57

• No, there is a remainder

What is the remainder when f(x) is divided by (x + 4)? ​ (a)  

Is (x + 4) a factor of f(x)? ​ (b)  

• r=-10

• No, there is a remainder

• Yes, the remainder is 0

• r=12

• r=-6

• r=42

• r=-42

• r=38

• r=0

• r=10

What are the three factors for
 (x³ +7x² +7x -15) ÷ (x -1)?

• (x-1) (x+5) (x-3) 

• (x-1) (x+5) (x + 3) 

• (x+1) (x+5) (x-3) 

• (x+1) (x-5) (x-3)