Apply division algorithm to find the quotient q(x) and remainder r(x) when f(x) = x³ − 6x² + 11x − 6 is divided by g(x) = x² + x + 1

Video Explanation

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Solution

Given:

f(x) = x³ − 6x² + 11x − 6
g(x) = x² + 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)

x³ − 6x² + 11x − 6 ÷ (x² + x + 1)

First term:

x³ ÷ x² = x

Multiply:

x(x² + x + 1) = x³ + x² + x

Subtract:

(x³ − 6x² + 11x) − (x³ + x² + x)

= −7x² + 10x

Bring down −6:

−7x² + 10x − 6

Next term:

−7x² ÷ x² = −7

Multiply:

−7(x² + x + 1) = −7x² − 7x − 7

Subtract:

(−7x² + 10x − 6) − (−7x² − 7x − 7)

= 17x + 1

Step 3: Identify Quotient and Remainder

Quotient, q(x) = x − 7

Remainder, r(x) = 17x + 1

Final Answer

Quotient: q(x) = x − 7
Remainder: r(x) = 17x + 1

Conclusion

Thus, by applying the division algorithm, when f(x) = x³ − 6x² + 11x − 6 is divided by g(x) = x² + x + 1, the quotient is x − 7 and the remainder is 17x + 1.