Check whether g(x) = x³ − 3x + 1 is a factor of f(x) = x⁵ − 4x³ + x² + 3x + 1 using division algorithm

Video Explanation

Watch the video explanation below:

Solution

Given:

g(x) = x³ − 3x + 1
f(x) = x⁵ − 4x³ + x² + 3x + 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)

x⁵ − 4x³ + x² + 3x + 1 ÷ (x³ − 3x + 1)

First term:

x⁵ ÷ x³ = x²

Multiply:

x²(x³ − 3x + 1) = x⁵ − 3x³ + x²

Subtract:

(x⁵ − 4x³ + x²) − (x⁵ − 3x³ + x²)

= −x³

Bring down +3x + 1:

−x³ + 3x + 1

Next term:

−x³ ÷ x³ = −1

Multiply:

−1(x³ − 3x + 1) = −x³ + 3x − 1

Subtract:

(−x³ + 3x + 1) − (−x³ + 3x − 1)

= 2

Step 3: Identify Quotient and Remainder

Quotient, q(x) = x² − 1

Remainder, r(x) = 2

Final Answer

Since the remainder is not zero, g(x) = x³ − 3x + 1 is not a factor of f(x).

Conclusion

Thus, by applying the division algorithm, we conclude that g(x) = x³ − 3x + 1 is not a factor of the polynomial f(x) = x⁵ − 4x³ + x² + 3x + 1.