Answer :

Let, f (x) = 2*x*^{4}-7*x*^{3}-13*x*^{2}+63*x*-45

The factors of the constant term – 45 are

The factor of the coefficient of x^{4} is 2. Hence, possible rational roots of f (x) are:

We have,

f (1) = 2 (1)^{4} – 7 (1)^{3} – 13 (1)^{2} + 63 (1) – 45

= 2 – 7 – 13 + 63 – 45

= 0

And,

f (3) = 2 (3)^{4} – 7 (3)^{3} – 13 (3)^{2} + 63 (3) – 45

= 162 – 189 – 117 + 189 – 45

= 0

So, (x – 1) and (x + 3) are the factors of f (x)

(x – 1) (x + 3) is also a factor of f (x)

Let us now divide

f (x) = 2*x*^{4}-7*x*^{3}-13*x*^{2}+63*x*-45 by (x^{2} – 4x + 3) to get the other factors of f (x)

Using long division method, we get

2*x*^{4}-7*x*^{3}-13*x*^{2}+63*x*-45 = (x^{2} – 4x + 3) (2x^{2} + x – 15)

2*x*^{4}-7*x*^{3}-13*x*^{2}+63*x*-45 = (x – 1) (x – 3) (2x^{2} + x – 15)

Now,

2x^{2} + x – 15 = 2x^{2} + 6x – 5x – 15

= 2x (x + 3) – 5 (x + 3)

= (2x – 5) (x + 3)

Hence, 2*x*^{4}-7*x*^{3}-13*x*^{2}+63*x*-45 = (x – 1) (x – 3) (x + 3) (2x – 5)

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