find the value of a&b if (x^2-5x+4) is a factor of the polynomial 2x^3+ax^2+bx-4. Express in factored form.?

Let f(x) = 2x³ + ax² + bx - 4
x² - 5x + 4 = (x - 1)(x - 4)
Hence, x - 1 and x - 4 are factors of f(x).

x - 1 is a factor of f(x):
By Factor Theorem, f(1) = 0
2(1)³ + a(1)² + b(1) - 4 = 0
a + b = 2 …… [1]

x - 4 is a factor of f(x):
By Factor Theorem, f(4) = 0
2(4)³ + a(4)² + b(4) - 4 = 0
16a + 4b = -124
4a + b = -31 …… [2]

[2] - [1]:
3a = -33
a = -11

Plug a = -11 into [1]:
(-11) + b = 2
b =13

f(x)
= 2x³ - 11x² + 13x - 4
= 2x³ - 2x² - 9x² + 9x + 4x - 4
= (2x³ - 2x²) - (9x² - 9x) + (4x - 4)
= 2x²(x - 1) - 9x(x - 1) + 4(x - 1)
= (x - 1)(2x² - 9x + 4)
= (x - 1)(x - 4)(2x - 1)

First, notice that x^2-5x+4 can be written as (x - 1)(x - 4), thus
x=1 & x=4 are factors. 

Therefore, f(x) = 2x^3+ax^2+bx-4 has factors 1 & 4. IOW,

f(1) = 0 & f(4) = 0. Setting this up will give you two (linear) equations with he two unknowns a & b. Just solve that linear system. Done!

Show your steps here if need be and we can further work it through with you.  


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