Study Mathematics notes on Polynomial Division and Completing the Square for Secondary. Useful for revision, homework, and exam preparation in Zimbabwe.
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Dividing Polynomials.
Example.
Find the quotent and remainder when the polynomal
$4 x^{2}+4 x-5$ is divided by $x+3$.
\[
\begin{array}{r}
\frac{x+1}{x+3 \sqrt{x^{2}+4 x-5}}
\left(x^{2}+3 x\right)
x-5
(x-3)
-8
\end{array}
\]
Quohent $=x+1$
Remainder $=-8$.
Find the quotient and remainder.
1. $6 x^{2}-7 x+5$ is divided by $2 x-3$.
\[
\begin{array}{r}
3 x+1
2 x-3 \sqrt{6 x^{2}-7 x+5}
-\left(6 x^{2}-9 x\right)
2 x+5
-(2 x-3)
8
\end{array}
\]
\[
\begin{array}{l}
\text { Quotent }=3 x+1
\text { Remainder }=8 .
\end{array}
\]
COMPLETING THE SILURE
Example.
Solve by completing the square.
\[
5 x^{2}-6 x-2=0
\]
We first make the coefficient of $x^{2} .1$ by direding throut by 5
\[
x^{2}-\frac{6}{5} x-\frac{2}{5}=0
\]
Complete the square by adding half of the coefficient of $x$ to both sides.
\[
\begin{aligned}
x^{2}-\frac{6}{5} x | =\frac{2}{5} .
x^{2}-\frac{6}{5} x+\left(\frac{3}{5}\right)^{2} | =\frac{2}{5}+\left(\frac{3}{5}\right)^{2} | =\frac{19}{25} .
\end{aligned}
\]
Then se foctorive the left Laudode
\[
\begin{array}{l}
\left(x-\frac{3}{5}\right)^{2}=\sqrt{\frac{19}{25}}
\left(x-\frac{3}{5}\right)= \pm \sqrt{\frac{19}{25}}
x=\frac{3}{5}+\frac{\sqrt{19}}{5} \text { or } \frac{3}{5}-\frac{\sqrt{19}}{5}
x=\frac{3+\sqrt{19}}{5} \text { or } \frac{3-\sqrt{19}}{5}
\end{array}
\]
2) Solve $x^{2}+3 x-1=0$.
\[
x^{2}+\left(\frac{3^{2}}{2}\right)^{2}=1+\left(\frac{3}{2}\right)^{2}
\]
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