Question

Using division of polynomials state whether

(i) x + 6 is a factor of x² - x - 42

(ii) 4x -1 is a factor of 4x² - 13x - 12

(iii) 2y - 5 is a factor of 4y⁴ - 10y³ - 10y² + 30y - 15

(iv) 3y² + 5 is a factor of 6y⁵ + 15y⁴ + 16y³ + 4y² + 10y - 35

(v) z² + 3 is a factor of z⁵ - 9z

(vi) 2x² - x + 3 is a factor of 6x⁵ - x⁴ + 4x³ - 5x² - x - 15

Answer 1

(i) x + 6 is a factor of x² - x - 42

Quotient: x- 7

Remainder: 0

Since remainder is 0 therefore (x + 6) is a factor of x² - x - 42

(ii) 4x -1 is a factor of 4x² - 13x - 12

Quotient: x - 3

Remainder: 15

Since remainder is 15 therefore is NOT a factor of 4x² - 13x - 12

(iii) 2y - 5 is a factor of 4y⁴ - 10y³ - 10y² + 30y - 15

Quotient: 2y³ - 5y + \(\frac{5}{2}\)

Remainder: -\(\frac{5}{2}\)

Since remainder is -\(\frac{5}{2}\) therefore (2y - 5) is NOT a factor of 4y⁴ - 10y³ - 10y² + 30y - 15

(iv) 3y² + 5 is a factor of 6y⁵ + 15y⁴ + 16y³ + 4y² + 10y - 35

Quotient: 2y³ + 2y + \(\frac{4}{3}\)

Remainder: - \(\frac{125}{3}\)

Since remainder is - \(\frac{125}{3}\) therefore (3y² + 5)  is NOT a factor of 6y⁵ + 15y⁴ + 16y³ + 4y² + 10y - 35

(v) z² + 3 is a factor of z⁵ - 9z

Quotient: z³ - 3z

Remainder: 0

Since remainder is 0 therefore (z³ - 3z) is a factor of (z⁵ - 9z)

(vi) 2x² - x + 3 is a factor of 6x⁵ - x⁴ + 4x³ - 5x² - x - 15

Quotient: 3x³ + x² - 2x - 3

Remainder: 2x² - x + 3

Since remainder is 2x - 6 therefore (3y² + 5) is NOT a factor of 6y⁵ + 15y⁴ + 16y³ + 4y² + 10y - 35