Solving Polynomial Remainder Theorem and Transformations

Solving Polynomials Using the Remainder Theorem

We are given the function:

f(x) = mx³ − 3x² + nx + 2

• When divided by (x + 3), the remainder is −3.
• When divided by (x − 2), the remainder is −4.

By the Remainder Theorem, if f(x) is divided by (x − a), the remainder is f(a).

Step 1: Substitute x = −3

f(−3) = m(−3)³ − 3(−3)² + n(−3) + 2

= −27m − 27 − 3n + 2

= −27m − 3n − 25

Given remainder −3:

−27m − 3n − 25 = −3

−27m − 3n = 22

27m + 3n = −22 (Equation 1)

Step 2: Substitute x = 2

f(2) = m(2)³ − 3(2)² + n(2) + 2

= 8m − 12 + 2n + 2

= 8m + 2n − 10

Given remainder −4:

8m + 2n − 10 = −4

8m + 2n = 6 (Equation 2)

Step 3: Solve the System of Equations

Multiply Equation 2 by 3 → 24m + 6n = 18

Multiply Equation 1 by 2 → 54m + 6n = −44

Subtracting the equations:

(54m + 6n) − (24m + 6n) = −44 − 18

30m = −62

m = −62 / 30 = −31 / 15

Step 4: Solve for n

Substitute m = −31/15 into Equation 2:

8(−31/15) + 2n = 6

−248/15 + 2n = 6

2n = 6 + 248/15 = 338 / 15

n = 169 / 15

Function Transformation Parameters

The general transformation formula is: y = a f(k(x − d)) + c

• a: Vertical stretch (a > 1), vertical compression (0 < a < 1), or reflection in the x-axis (a < 0).
• k: Horizontal stretch (k < 1), horizontal compression (k > 1), or reflection in the y-axis (k < 0).
• d: Horizontal translation (positive moves right, negative moves left).
• c: Vertical translation (positive moves up, negative moves down).

Determining X-Intercepts

Determine the x-intercepts of the following functions (rounded to two decimal places):

(a) y = 3(x − 2)⁴ − 5

0 = 3(x − 2)⁴ − 5 → 5/3 = (x − 2)⁴

x − 2 = ± ⁴√(5/3) ≈ ± 1.14

x = 3.14 or x = 0.86

(b) y = (x + 3)³ − 5

0 = (x + 3)³ − 5 → 5 = (x + 3)³

x + 3 = ³√5 ≈ 1.71

x = −1.29

Mapping Coordinates

Given a function reflected in the x-axis, horizontally compressed by 1/3, translated 5 units left, and 4 units up, find the original coordinates for point (−6, 31).

Mapping formula: (x₂, y₂) = (x₁/k + d, a·y₁ + c)

Where a = −1, k = 3, d = −5, c = 4:

−6 = x₁/3 − 5 → x₁/3 = −1 → x₁ = −3

31 = −y₁ + 4 → −y₁ = 27 → y₁ = −27

Original coordinates: (−3, −27)