Previous Class Test question - complex numbers

 

Complex numbers example - Algebra Notes Chapter 3, Q47, Q48

•  Complex numbers example - Algebra Notes Chapter 3, Q47ab, Q48

Complex polynomial factorise fully (Mobius question 2023 Oct) - Like Q65 in Algebra Notes Ch3

 

Basic complex numbers calculations

 

Basic calculation with complex numbers, example (class test 2014)

 

Basic complex number calculations (class test 2014)

 

Find roots and factorise $z^6+64$ (class test 2014)

 

Factorise into real linear and quadratic factors, $p(z)=z^5-32$, (Alg Class test 2014) Q3

 

Find the complex square roots (Alg Class test 2014)

 

Factorise $p(z)=z^5+z^4-8z^3-28z^2+16z+96$ into linear factors.

 The original question was that given $-2-2i$ is a root and that $3$ other roots were real, factorise into linear factors. Since $p(z)$ has real coefficients this means the conjugate root is also a root, so both $-2-2i$ and $-2+2i$ are roots. 

Here we assume that there are $3$ real roots and go from there.

Principal Arguments of z and $-z$ and also $z-w$ and $w-z$.

 A student asked me about these.

Try them before you look at the answer!!

Q3

Q4

Factorise $p(z)=z^5-5z^4+10z^3-10z^2-11z+15$ given that $z=1-2i$ is a root and that three roots are integers.(mobius)

 

This is the factorisation of $p(z)$ into real linear factors and real quadratic factors.

ANSWER: The answer to the original question is that the complex linear factors of $p(z)$ are $$(z-1),(z+1),(z-3),(z-1+2i)(z-1-2i)\ .$$

Find mod and Arg of a power of a complex number in mod-arg form. Example

 

Ch3 Q61 Factorise $z^6+8$ into both linear and quadratic factors.

 

Express $\cos^5t$ in terms of $\cos 5t,\cos3t,\cos t$ and express $\sin^4t$ in terms of $\cos 4t, \cos 2t$

• Other examples like this one are contained in the Algebra Notes Ch3 page 98 Example 3 and 4 ! Please also read this book, as it has many useful examples!!
• RTB - Read the Book 😎

Sketch the ray $\text{Arg}\ (z-(3+2i))=\dfrac{\pi}{4}$

  

Find the square roots of $i$ in two ways, Cartesian and Mod-Arg way.

  

Challenge:Complex reducible to quadratic equation: Solve $$z^4+2\left(a^2-b^2\right)z^2+\left(a^2+b^2\right)^2=0$$ where $a,b\in\mathbb{R}$

Solve $$z^4+2\left(a^2-b^2\right)z^2+\left(a^2+b^2\right)^2=0$$ where $a,b$ are real numbers.

Hint 1 : Put $w=z^2$ and find $w$ first.  Then for each $w$, solve $z^2=w$.

Hint 2: see other posts on this label.

Answer: $z=\pm (a+ib), \pm (a-ib)$

Complex reducible to quadratics equation: Solve $$z^4+16z^2+100=0$$ .

Question: Solve $$z^4+16z^2+100=0$$ .

Show the answers are $$z=\pm(1+3i), \pm(1-3i)\ \ .$$

Complex reducible to quadratic equation: Solve $$z^4+12z^2+169=0$$

 

Try solving this one now

$$z^4+16z^2+100=0$$ and show the solutions are ....https://dipa2023t2.blogspot.com/2023/06/complex-reducible-to-quadratics.html