Find Polynomial coefficients

Question

How can I find the real coefficients a,b if the polynomial

$ P(X)=X^4-5X^3+8X^2+aX+b $ is divisible by $ Q(X)=(X-1)^2 $

So if 1 is a solution, I get a+b=-4; where should i look for other roots?

try: divide by x-1 first, and then divide the rest by x-1 again — TYZ, May 02 '14 at 15:10

score 1 · Accepted Answer · answered May 02 '14 at 15:12

1

Just compute $Q(x)\cdot(x^2+cx+d)-P(x)=0$ and compare coefficients. This gives immediately $a=-5$, $b=1$, $c=-3$ and $d=1$.

answered May 02 '14 at 15:12

Dietrich Burde

130,978

score 1 · Answer 2 · answered May 02 '14 at 15:13

1

If (x-1)^2 divides p(x) it is also root of p'(x) . Therefore 1 is a root of the functions differential by which u can find a then b

answered May 02 '14 at 15:13

avz2611

3,658

score 0 · Answer 3 · answered May 09 '16 at 17:34

Using synthetic division:

\begin{array}{rrrrrrr} & | & 1 & -5 & 8 & a & b \\ 1 & | & 0 & 1 & -4 & 4 & a+4 \\ & + & --- & --- & --- & --- & ----- \\ & & 1 & -4 & 4 & a+4 & a+b+4 \\ \end{array}

So $a+b+4 = 0$.

Dividing again by $(x-1)$ :

\begin{array}{rcrrrr} & | & 1 & -4 & 4 & a+4 \\ 1 & | & 0 & 1 & -3 & 1 \\ & + & --- & --- & --- & --- \\ & & 1 & -3 & 1 & a+5 \\ \end{array}

So $a+5= 0$, hence $a = -5$ and $b=1$.

Find Polynomial coefficients

3 Answers3