Question

Using factor theorem, factorize each of the following polynomial:
2x⁴-7x³-13x²+63x-45

Answer 1

2x⁴ - 7x³ - 13x² + 63x - 45 =  (x - 1) (x - 3) (x + 3)(2x - 5)

Step-by-step explanation:

f(x) = 2x⁴ - 7x³ - 13x² + 63x - 45

x = 1

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

= 2 - 7 - 13 + 63 - 45

= 0

=> x = 1 is a factor

(x - 1) is a factor

                    2x³ - 5x² - 18x + 45

x - 1         _|     2x⁴ - 7x³ - 13x² + 63x - 45

                      2x⁴  - 2x³

                    ___________

                              -5x³ - 13x² + 63x - 45

                              -5x³  + 5x²

                              _______________

                                       -18x² + 63x - 45

                                        -18x²  + 18x

                                        ______________

                                                     45x  - 45

                                                      45x - 45

                                                     _______

                                                           0

2x⁴ - 7x³ - 13x² + 63x - 45 = (x - 1)  (2x³ - 5x² - 18x + 45)

Similarly

x = 3  => 2x³ - 5x² - 18x + 45 = 0

x - 3 is factor

=> 2x⁴ - 7x³ - 13x² + 63x - 45 = (x - 1) (x - 3) (2x² + x - 15)

(x - 1) (x - 3) (2x² + x - 15)

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

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

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

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

2x⁴ - 7x³ - 13x² + 63x - 45 =  (x - 1) (x - 3) (x + 3)(2x - 5)

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Answer 2

$\huge\star\mathfrak\blue{{Answer:-}}$

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)