Question

Hi. I need help with these questions.See image for question.​

Answer 1

Step-by-step explanation:

Given

• f(x) = 4x³ + 3x² - 2x - 1

Divide it by the following:

(a) 2x + 1

• 4x³ + 3x² - 2x - 1 =
• (4x³ + 2x²) + (x² + 1/2x) - (5/2x + 5/4) + 1/4 =
• 2x²(2x+1) + 1/2x(2x + 1) - 5/4(2x + 1) + 1/4 =
• (2x + 1)(2x² + 1/2x - 5/4) + 1/4

Quotient =  2x² + 1/2x - 5/4

Remainder = 1/4

(b) 2x - 3

• 4x³ + 3x² - 2x - 1 =
• (4x³ - 6x²) + (9x² - 13.5x) + (11.5x - 17.25) + 16.25 =
• (2x -3)(2x² + 4.5x + 5.75) + 16.25

Quotient = 2x² + 4.5x + 5.75

Remainder = 16.25

(c) 4x - 1

• 4x³ + 3x² - 2x - 1 =
• (4x³ - x²) + (4x² - x) - (2x - 1/2) - 3/2 =
• (4x - 1)(x² + x - 1/2) - 3/2

Quotient = x² + x - 1/2

Remainder = - 3/2

(d) x + 2

• 4x³ + 3x² - 2x - 1 =
• (4x³ + 8x²) - (5x² + 10x) + (8x + 16) - 17 =
• (x + 2)(4x² - 5x + 8) - 17

Quotient = 4x² - 5x + 8

Remainder = - 17

Answer 2

Step-by-step explanation:

Given

f(x) = 4x³ + 3x² - 2x - 1

Divide it by the following:

(a) 2x + 1

4x³ + 3x² - 2x - 1 =

(4x³ + 2x²) + (x² + 1/2x) - (5/2x + 5/4) + 1/4 =

2x²(2x+1) + 1/2x(2x + 1) - 5/4(2x + 1) + 1/4 =

(2x + 1)(2x² + 1/2x - 5/4) + 1/4

Quotient =  2x² + 1/2x - 5/4

Remainder = 1/4

(b) 2x - 3

4x³ + 3x² - 2x - 1 =

(4x³ - 6x²) + (9x² - 13.5x) + (11.5x - 17.25) + 16.25 =

(2x -3)(2x² + 4.5x + 5.75) + 16.25

Quotient = 2x² + 4.5x + 5.75

Remainder = 16.25

(c) 4x - 1

4x³ + 3x² - 2x - 1 =

(4x³ - x²) + (4x² - x) - (2x - 1/2) - 3/2 =

(4x - 1)(x² + x - 1/2) - 3/2

Quotient = x² + x - 1/2

Remainder = - 3/2

(d) x + 2

4x³ + 3x² - 2x - 1 =

(4x³ + 8x²) - (5x² + 10x) + (8x + 16) - 17 =

(x + 2)(4x² - 5x + 8) - 17

Quotient = 4x² - 5x + 8

Remainder = - 17