Apply division algorithm to find the quotient q(x) and remainder r(x) when f(x) = 15x³ − 20x² + 13x − 12 is divided by g(x) = 2 − 2x + x²

Video Explanation

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Solution

Given:

f(x) = 15x³ − 20x² + 13x − 12
g(x) = 2 − 2x + x²

First arrange g(x) in descending powers of x:

g(x) = x² − 2x + 2

Step 1: Apply the Division Algorithm

According to the division algorithm,

f(x) = g(x) · q(x) + r(x)

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

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

15x³ − 20x² + 13x − 12 ÷ (x² − 2x + 2)

First term:

15x³ ÷ x² = 15x

Multiply:

15x(x² − 2x + 2) = 15x³ − 30x² + 30x

Subtract:

(15x³ − 20x² + 13x) − (15x³ − 30x² + 30x)

= 10x² − 17x

Bring down −12:

10x² − 17x − 12

Next term:

10x² ÷ x² = 10

Multiply:

10(x² − 2x + 2) = 10x² − 20x + 20

Subtract:

(10x² − 17x − 12) − (10x² − 20x + 20)

= 3x − 32

Step 3: Identify Quotient and Remainder

Quotient, q(x) = 15x + 10

Remainder, r(x) = 3x − 32

Final Answer

Quotient: q(x) = 15x + 10
Remainder: r(x) = 3x − 32

Conclusion

Thus, by applying the division algorithm, when f(x) = 15x³ − 20x² + 13x − 12 is divided by g(x) = 2 − 2x + x², the quotient is 15x + 10 and the remainder is 3x − 32.