Apply division algorithm to find the quotient q(x) and remainder r(x) when f(x) = 10x⁴ + 17x³ − 62x² + 30x − 3 is divided by g(x) = 2x² + 7x + 1

Video Explanation

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Solution

Given:

f(x) = 10x⁴ + 17x³ − 62x² + 30x − 3
g(x) = 2x² + 7x + 1

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)

10x⁴ + 17x³ − 62x² + 30x − 3 ÷ (2x² + 7x + 1)

First term:

10x⁴ ÷ 2x² = 5x²

Multiply:

5x²(2x² + 7x + 1) = 10x⁴ + 35x³ + 5x²

Subtract:

(10x⁴ + 17x³ − 62x²) − (10x⁴ + 35x³ + 5x²)

= −18x³ − 67x²

Bring down +30x:

−18x³ − 67x² + 30x

Next term:

−18x³ ÷ 2x² = −9x

Multiply:

−9x(2x² + 7x + 1) = −18x³ − 63x² − 9x

Subtract:

(−18x³ − 67x² + 30x) − (−18x³ − 63x² − 9x)

= −4x² + 39x

Bring down −3:

−4x² + 39x − 3

Next term:

−4x² ÷ 2x² = −2

Multiply:

−2(2x² + 7x + 1) = −4x² − 14x − 2

Subtract:

(−4x² + 39x − 3) − (−4x² − 14x − 2)

= 53x − 1

Step 3: Identify Quotient and Remainder

Quotient, q(x) = 5x² − 9x − 2

Remainder, r(x) = 53x − 1

Final Answer

Quotient: q(x) = 5x² − 9x − 2
Remainder: r(x) = 53x − 1

Conclusion

Thus, by applying the division algorithm, when f(x) = 10x⁴ + 17x³ − 62x² + 30x − 3 is divided by g(x) = 2x² + 7x + 1, the quotient is 5x² − 9x − 2 and the remainder is 53x − 1.