Apply division algorithm to find the quotient q(x) and remainder r(x) when f(x) = 4x³ + 8x² + 8x + 7 is divided by g(x) = 2x² − x + 1

Video Explanation

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Solution

Given:

f(x) = 4x³ + 8x² + 8x + 7
g(x) = 2x² − x + 1

Step 1: Apply the Division Algorithm

According to the division algorithm,

f(x) = g(x) · q(x) + r(x)

where degree of r(x) is less than degree of g(x).

Step 2: Divide f(x) by g(x)

4x³ + 8x² + 8x + 7 ÷ (2x² − x + 1)

First term:

4x³ ÷ 2x² = 2x

Multiply:

2x(2x² − x + 1) = 4x³ − 2x² + 2x

Subtract:

(4x³ + 8x² + 8x) − (4x³ − 2x² + 2x)

= 10x² + 6x

Bring down +7:

10x² + 6x + 7

Next term:

10x² ÷ 2x² = 5

Multiply:

5(2x² − x + 1) = 10x² − 5x + 5

Subtract:

(10x² + 6x + 7) − (10x² − 5x + 5)

= 11x + 2

Step 3: Identify Quotient and Remainder

Quotient, q(x) = 2x + 5

Remainder, r(x) = 11x + 2

Final Answer

Quotient: q(x) = 2x + 5
Remainder: r(x) = 11x + 2

Conclusion

Thus, by applying the division algorithm, when f(x) = 4x³ + 8x² + 8x + 7 is divided by g(x) = 2x² − x + 1, the quotient is 2x + 5 and the remainder is 11x + 2.