Unit 2 - Polynomial Functions and Equations

Dividing polynomials

100

State the degree of the quotient for the following division statement: 

(x⁴ - 15x³ + 2x² + 12x - 10) ÷ (x² - 4) 

2

200

Solve for the quotient using synthetic division: 

(6x³ - 2x - 15x² + 5) ÷ (2x - 5) 

= 3x² - 1

300

Calculate the quotient using long division: 

(x⁴ + 3x³ - 2x² + 5x - 1) ÷ (x² + 7)

x² + 3x - 9

Remainder: -16x + 62

Factor Theorem

100

Which of the following functions are divisible by (x - 1)?

a) f(x) = x⁴ - 15x³ + 2x² + 12x - 10

b) g(x) = 5x³ - 4x² + 3x - 4

c) h(x) = x⁴ - 7x³ + 2x² + 9x

d) j(x) = x³ - 1

B and D

200

State the remainder of the following equation: 

(x⁴ - 5x² + 4) ÷ (x + 2)

Remainder = 0 

300

Determine whether (2x - 5) is a factor of (2x⁴ - 7x³ - 13x² + 63x - 45)

Yes - the remainder is 0

Polynomial Equations

100

State the number of zeroes in the following function: 

f(x) = (x² - x- 12)(3x) 

3 zeroes

200

State the zeroes of the following function: 

f(x) = -3x³ (2x+4) (x² - 25)

x = -5, -2, 0, 5

300

Determine the roots of the equation:  

4x⁴ - 4x³ - 51x² + 106x = 40

x = -4, 1/2, 2, 5/2

Polynomial Inequalities

100

Solve the inequality algebraically: 

2x - 1 ≤ 13

x ∈ ( -∞ , 7]

200

Solve the inequality: 

2x(x+4) - 3(x+4) ≤ 0 

x ∈ [-4 , 3/2)

300

Solve the following inequality algebraically and write the answer using interval notation: 

3x³ - 3x² - 2x ≤ 2x³ - x² + x 

x ∈ ( -∞ , -1) U [0, 3]