Polynomials (Manipulation and Zeros)
NZ Level 7 (NZC) Level 2 (NCEA)
Further applications of remainder and factor theorem

## Interactive practice questions

The polynomials $4x^2-7x-15$4x27x15 and $5x^2+13x+k$5x2+13x+k have a common factor of $x+p$x+p, where $p$p is an integer.

a

Using the fact that $x+p$x+p is a factor of $4x^2-7x-15$4x27x15, solve for the value of $p$p.

b

Using the fact that $x+p$x+p is a factor of $5x^2+13x+k$5x2+13x+k, solve for $k$k.

Easy
Approx 7 minutes
Consider the polynomial $x^{99}+1$x99+1.
The polynomials $P\left(x\right)=x^3+4x^2-5x+n$P(x)=x3+4x25x+n and $Q\left(x\right)=x^3+2x+17$Q(x)=x3+2x+17 leave the same remainder when divided by $x+1$x+1.
Solve for the value of $n$n.
Consider the sequence: $2^5-2$252, $3^5-3$353, $4^5-4$454, $\text{. . .}$. . ., $x^5-x$x5x.