MAT102 Tutorial W2

• For any function $f$, $r$ is a root if $f(r)=0$

  • Definition

• If we have a polynomial $p(x)=a{x}^2+bx+c$ it must follow the cutsy rule $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

  • Theorem

• For any polynomial $p(x)=a{x}^2+bx+c$

  • If discriminant in quadratic formula is less than 0, no roots, = 0, one root and greater than 0, 2 roots.
  • Corollary

• Show that $x^2+x+1$ has no roots

  • $a=1$
  • $b=1$
  • $c=1$
  • $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
  • $x=\frac{-b\pm \sqrt{1^2-4}}{2a}$
  • $x=\frac{-b\pm \sqrt{1-4}}{2a}$
  • $x=\frac{-b\pm \sqrt{-3}}{2a}$
  • Since the discriminant is $<0$, there is no roots to this quadratic equation.

• $2=1$

  • $a=b$
  • $a^2=ab$
  • $a^2-b^2=ab-b^2$
  • $(a+b)(a-b)=b(a-b)$
  • $a^2-b^2=ab-b^2$
  • $a^2=ab$

• Every perfect square is even

  • $a^2=2b$
  • $b\in R$

• The difference of any odd and any even number is odd.

  • $n\in Z$
  • $2n$ is even
  • $2n+1$ is odd
  • $2n+1-2n=1$

• Prove that is $A\cup B=A$ then $B\subseteq A$