04 Polynomial division and remainders · g10-04-01 · 5–10 min
What does division preserve?
How can the quotient and remainder reconstruct f?
Predict: For a cubic divided by x−a, predict the degree of the quotient and the shape of the remainder.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
x² coefficient b = -2; x coefficient c = 0; Constant d = 3; Divisor x−a = 2 | |
Case 2
x² coefficient b = 0; x coefficient c = -1; Constant d = 0; Divisor x−a = 1 | |
Case 3
x² coefficient b = 2; x coefficient c = -3; Constant d = 4; Divisor x−a = 0 | |
Revise: My claim needs the condition… because…
Exit: Divide x²+1 by x−1.
04 Polynomial division and remainders · g10-04-02 · 5–10 min
Where did the missing terms go?
Can missing coefficients be omitted in synthetic division?
Predict: For x⁴+bx²+c, predict the five coefficient slots before dividing by x−a.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
x² coefficient b = -3; Constant c = 2; Synthetic multiplier a = 1 | |
Case 2
x² coefficient b = 0; Constant c = 0; Synthetic multiplier a = 2 | |
Case 3
x² coefficient b = 2; Constant c = -1; Synthetic multiplier a = 0 | |
Revise: My claim needs the condition… because…
Exit: List the coefficient row for x⁴−3x²+2.
04 Polynomial division and remainders · g10-04-03 · 5–10 min
Read a remainder from one point
Why is the remainder f(a)?
Predict: Divide f(x)=x³+bx+c by x−a; predict which graph input reveals the remainder.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
x coefficient b = -2; Constant c = 3; Divisor zero a = 2 | |
Case 2
x coefficient b = 0; Constant c = 1; Divisor zero a = -2 | |
Case 3
x coefficient b = -1; Constant c = 0; Divisor zero a = 1 | |
Revise: My claim needs the condition… because…
Exit: Find the remainder of x³+1 divided by x+2.
04 Polynomial division and remainders · g10-04-04 · 5–10 min
Design a factor by changing one coefficient
What constant makes x−a a factor?
Predict: For x³+kx+c, predict the required c before comparing the original and repaired graphs.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Required root a = 1; x coefficient k = 2; Original constant c = 0 | |
Case 2
Required root a = -1; x coefficient k = 2; Original constant c = 3 | |
Case 3
Required root a = 0; x coefficient k = -2; Original constant c = 0 | |
Revise: My claim needs the condition… because…
Exit: Find c so x+1 divides x³+2x+c.
