Question

Use the division algorithm to divide f(x)=x⁵+x³+x+1 by g(x)=3x²+5 x+2. Write your answer in the form f(x)=q(x) g(x)+r(x) .

Answer 1

To divide the polynomial f(x) by g(x), we perform polynomial long division until the remainder r(x) has a lesser degree than g(x). The result is expressed in the form f(x) = q(x)g(x) + r(x), where q(x) is the quotient and r(x) is the remainder. f(x)= 1/3x3 - 2/9x2 + 14/27x - 15/81 + 314/27 ÷ (3x2 +5x+2)

Step-by-step explanation:

Sure, let's go through the polynomial long division step by step:

Divide the leading term of f(x) by the leading term of g(x):

x5/ 3x2 = 1/3x3

Multiply g(x) by the result from step 1 and subtract it from f(x):

x5 + x3 + x+1− ( 1/3x3×(3x2 +5x+2))

This simplifies to:

x5 + x3 + x+1− (x5 +0x4 + 2/3x3)

After simplification:

2/3x3 - 2/9x2 + x + 1

Repeat the process. Divide the leading term of the result from step 2 by the leading term of g(x):

2/3x3 ÷ 3x2 = 2/9x

Multiply g(x) by the result from step 3 and subtract it from the result of step 2:

2/3x3 − 2/9x2 +x+1 − ( 2/9x × (3x2 +5x+2))

This simplifies to:

2/3x3 − 2/9x2 +x+1 − (2/3x3 + 10/9x2 + 4/9x)

After simplification:

14/27x − 15/81

Repeat the process until the degree of the result is less than the degree of g(x).

The final result is:

f(x)= 1/3x3 - 2/9x2 + 14/27x - 15/81 + 314/27 ÷ (3x2 +5x+2)