Questions

1. What is the remainder of

$$\frac{x^{28}-2x^5+3}{x+1}\text{\hspace{0.17em}?}$$

2. Find the value of k so that (x-3) is a factor of:

$$x^4+k{x}^2+9x-27.$$

Solutions

1) By the Remainder Theorem, remainder on division by

$x+1$

is

$P(-1)$

:

$$(-1)^{28}-2(-1)^5+3=1+2+3=6.$$

2)

$x-3$

is a factor iff

$P(3)=0$

:

$$3^4+k\cdot 3^2+9\cdot 3-27=81+9k+27-27=81+9k.$$

Set

$81+9k=0\Rightarrow k=-9$

.