Question

2 x^4 "-13x^3" +19x^2 +5x +1 = (x-4)(Ax+1)(x^2 +Bx +1) +C is true for all the values of x ; Find the values of A, B and C

Answer 1

A=2

B=-3

C=-5

Explanation:

The highiest coefficient in degree of the polynomial is 2 on the left and 1*A*1,

A=2

Let's make the synthetical division:

`\boxed{A=2}\\\begin{array}c&x^4&x^3&x^2&x&1\\---&---&---&---&---&---\\&2&-13&19&5&1\\x=4&&8&-20&-4&4\\&2&-5&-1&1&-5\Longrightarrow \boxed{C=-5}\\x=-(1)/(2)&-1&3&-1&\\2&-6&2&&\\---&---&---&---&---&---\\\end {array}\\\\\\\boxed{2x^4-13x^3+19x^2+5x+1=(x-4)(2x+1)(x^2-3x+1)-5}\\\\\\\boxed{B=-3}\\`