# Polynomial

• Dec 16th 2012, 06:25 AM
notorious96
Polynomial
Question>> the polynomial x^4+5x+a is denoted by p(x). It is given that x^2-x+3 is a factor of p(x).
part (1) Find the value of a in p(x).

Can somebody please reply to this quickly. Thank you(Talking)
• Dec 16th 2012, 06:41 AM
jakncoke
Re: Polynomial
\$\displaystyle (x^2 - x + 3)(a_2x^2 + a_1x + a_0) = x^4 + 5x + a\$ so [tex] we get \$\displaystyle a_2x^4 + a_1x^3 + a_0x^2 - a_2x^3 - a_1 x^2 - a_0x + a_2*3*x^2 + 3*a_1x + 3a_0= a_2x^4 + (a_1 - a_2)x^3 + (a_0 - a_1 + 3a_2)x^2 + (3a_1 - a_0)x + 3a_0 = x^4 + 5x + a\$ which means a_2 = 1. So \$\displaystyle a_2x^4 + (a_1 - 1)x^3 + (a_0 - a_1 + 3)x^2 + (3a_1 - a_0)x + 3a_0\$ and since \$\displaystyle a_1 - 1 = 0 \$ \$\displaystyle a_1 = 1 \$. So now you got \$\displaystyle a_2x^4 + (a_0 + 2)x^2 + (3 - a_0)x + 3a_0\$ and since \$\displaystyle a_0 + 2 = 0 \$ \$\displaystyle a_0 = -2 \$ so again \$\displaystyle x^4 + (5)x -6\$ so a = -6.
• Dec 16th 2012, 07:20 AM
skeeter
Re: Polynomial
Code:

```................. x^2  +  x  - 2 .................----------------------------- x^2 - x + 3 | x^4 + 0x^3 + 0x^2 + 5x + a ............  x^4 -  x^3  + 3x^2 ..............----------------------------- ....................x^3  - 3x^2 + 5x + a ....................x^3  -  x^2 + 3x ....................--------------------- ........................- 2x^2 + 2x + a ........................- 2x^2 + 2x - 6 ........................---------------- ......................................a+6```
remainder, a+6 = 0

a = -6
• Dec 16th 2012, 07:25 AM
notorious96
Re: Polynomial
thanks jakncoke. Thankyou skeeter, i was also using the same method but my remainder included the variable 'x' so i thought it was wrong. Found my mistake though, thanks once again.