Using factor theorem, factories : p(x) = 2x4 - 7x3 - 13x2 + 63x - 4

1.	Using factor theorem, factories : p(x) = 2x4 - 7x3 - 13x2 + 63x - 45
Answer» 45 ⇒ ±1,±3,±5,±9,±15,±45 if we put x = 1 in p(x) p(1) = 2(1)⁴ - 7(1)³ - 13(1)² + 63(1) - 45 2 - 7 - 13 + 63 - 45 = 65 - 65 = 0 ∴ x = 1 or x - 1 is a factor of p(x). Similarly, if we put x = 3 in p(x) p(3) = 2(3)⁴ - 7(3)³ - 13(3)² + 63(3) - 45 162 - 189 - 117 + 189 - 45 = 162 - 162 = 0 Hence, x = 3 or x - 3 = 0 is the factor of p(x). p(x) = 2x⁴ - 7x³ - 13x² + 63x - 45 ∴ p(x) = 2x³ (x - 1) -5x² (x - 1) - 18(x - 1) + 45(x - 1) 2x⁴ - 2x³ (x - 1) - 5x² - 18x² + 18x + 45x - 54 ⇒ p(x) = (x - 1)(2x³ - 5x² - 18x + 45) ⇒ p(x) = (x - 1)(2x³ - 5x² - 18x + 45) ⇒ p(x) = (x - 1)[2x² (x - 3) + x(x - 3) - 15(x - 3)] ⇒ p(x) = (x - 1)[2x³ - 6x² + x² - 3x - 15x + 45] ⇒ p(x) = (x - 1)(x - 3)(2x² + x - 15) ⇒ p(x) = (x - 1)(x - 3)(2x² + 6x - 5x - 15) ⇒ p(x) = (x - 1)(x - 3)[2x(x + 3) - 5(x + 3)] ⇒ p(x) = (x - 1)(x - 3)(x + 3)(2x - 5)

Using factor theorem, factories : p(x) = 2x4 - 7x3 - 13x2 + 63x - 45

Answer»

45 ⇒ ±1,±3,±5,±9,±15,±45

if we put x = 1 in p(x)

p(1) = 2(1)⁴ - 7(1)³ - 13(1)² + 63(1) - 45

2 - 7 - 13 + 63 - 45 = 65 - 65 = 0

∴ x = 1 or x - 1 is a factor of p(x).

Similarly, if we put x = 3 in p(x)

p(3) = 2(3)⁴ - 7(3)³ - 13(3)² + 63(3) - 45

162 - 189 - 117 + 189 - 45 = 162 - 162 = 0

Hence, x = 3 or x - 3 = 0 is the factor of p(x).

p(x) = 2x⁴ - 7x³ - 13x² + 63x - 45

∴ p(x) = 2x³ (x - 1) -5x² (x - 1) - 18(x - 1) + 45(x - 1)

2x⁴ - 2x³ (x - 1) - 5x² - 18x² + 18x + 45x - 54

⇒ p(x) = (x - 1)(2x³ - 5x² - 18x + 45)

⇒ p(x) = (x - 1)[2x² (x - 3) + x(x - 3) - 15(x - 3)]

⇒ p(x) = (x - 1)[2x³ - 6x² + x² - 3x - 15x + 45]

⇒ p(x) = (x - 1)(x - 3)(2x² + x - 15)

⇒ p(x) = (x - 1)(x - 3)(2x² + 6x - 5x - 15)

⇒ p(x) = (x - 1)(x - 3)[2x(x + 3) - 5(x + 3)]

⇒ p(x) = (x - 1)(x - 3)(x + 3)(2x - 5)