Question

What should be subtracted from the
polynomial 5x⁴- 15 x³+ 2x² so that 5x³-1
becomes its
factor ?​

Answer 1

Step-by-step explanation:

Let k must be subtracted from 2x^4–4x^3+4x^2–4x+3. , therefore

2x^4 -4x^3+ 4x^2 -4x + 3 -k.

= 2x^2.(x^2+x+1) -2x^3 -2x^2–4x^3+4x^2–4x+3-k.

=2x^2.(x^2+x+1)-6x^3+2x^2–4x+3-k.

=2x^2.(x^2+x+1)-6x.(x^2+x+1)+6x^2+6x+2x^2–4x+3-k.

=2x^2.(x^2+x+1)-6x.(x^2+x+1)+8x^2+2x+3-k.

=2x^2.(x^2+x+1)-6x.(x^2+x+1)+8(x^2+x+1)-8x-8+2x+3-k.

=2x^2.(x^2+x+1)-6x.(x^2+x+1)+8.(x^2+x+1)-6x-5-k.

=(x^2+x+1)×(2x^2–6x+8) +(-6x-5-k).

= divisor × quotient. + remainder.

Remainder (-6x-5-k) should be zero.

or. -6x-5-k=0

or. k=-6x-5. Answer