What is the smallest number of real roots that a cubic , with real coefficients, can have?

The smallest number of real roots that a cubic , with real coefficients, can have is one . It is not possible for a cubic to have zero real roots .

True or False? A quartic , with real coefficients , can have at most two distinct non-real roots .

False. A quartic , with real coefficients , can have at most four distinct non-real roots .

How many cube roots does a complex number have?

A complex number has three cube roots .

True or False? The n th roots of a complex number form a regular polygon when plotted on an Argand diagram.

True. The n th roots of a complex number form a regular polygon when plotted on an Argand diagram.

IBMathsDPAnalysis & Approaches (AA)HLFlashcards1. Number & AlgebraFurther Complex Numbers

Further Complex Numbers (DP IB Analysis & Approaches (AA))

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FrontGeometry of Complex Numbers
What happens geometrically to a complex number, $z$ , when the complex number $a + b i$ is added to it?

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What happens geometrically to a complex number, $z$ , when the complex number $a + b i$ is added to it?
When the complex number $a + b i$ is added to $z$ , $z$ is translated on the Argand diagram by the vector $(\begin{matrix} a \\ b \end{matrix})$ .
What happens geometrically to a complex number, $z$ , when the complex number $a + b i$ is subtracted from it?
When the complex number $a + b i$ is subtracted from $z$ , $z$ is translated on the Argand diagram by the vector $(\begin{matrix} - a \\ - b \end{matrix})$ .
Describe the geometric relationship between the origin of an Argand diagram and the complex numbers $z$ , $w$ and $z + w$ .
The origin of an Argand diagram and the complex numbers $z$ , $w$ and $z + w$ form a parallelogram. $z + w$ is the vertex opposite the origin.
Describe the geometric relationship between the origin of an Argand diagram and the complex numbers $z$ , $w$ and $z - w$ .
The origin of an Argand diagram and the complex numbers $z$ , $w$ and $z - w$ form a parallelogram. $z - w$ is the vertex opposite the origin.
What two geometrical transformations happen to a complex number, $z$ , when it is multiplied by the complex number $w$ ?
When $z$ is multiplied by the complex number $w$ :
- it is rotated counter-clockwise by $\arg (w)$ ,
- it is stretched from the origin by scale factor $|w|$ .
What two geometrical transformations happen to a complex number, $z$ , when it is divided by the complex number $w$ ?
When $z$ is divided by the complex number $w$ :
- it is rotated clockwise by $\arg (w)$ ,
- it is stretched from the origin by scale factor $\frac{1}{|w|}$ .
Describe the geometrical relationship between $z$ and $z *$ .
On an Argand diagram, $z$ and $z *$ are reflections in the real axis.
What does a complex number look like when written in modulus-argument (polar) form?
A complex number that is written in modulus-argument (polar) form looks like $z = r (\cos θ + isin θ)$ where $r$ is the modulus and $θ$ is the argument.
What is denoted by $r cis (θ)$ ?
$r cis θ = r (\cos θ + isin θ)$ .
What is the complex conjugate of $r cis (θ)$ ?
The complex conjugate of $r cis (θ)$ is $r cis (- θ)$ .
True or False?
$2 (\cos (\frac{π}{3}) - isin (\frac{π}{3}))$ is written in modulus-argument (polar) form.
False.
$2 (\cos (\frac{π}{3}) - isin (\frac{π}{3}))$ is not written in modulus-argument (polar) form. There should be a "+" in front of $i$ .
Write the complex number $r (\cos θ - isin θ)$ in modulus-argument (polar) form.
The complex number $r (\cos θ - isin θ)$ in modulus-argument (polar) form is $r (\cos (- θ) + isin (- θ))$ .
If you know the modulus ( $r$ ) and argument ( $θ$ ) of a complex number, how can you find the real part?
If you know the modulus ( $r$ ) and argument ( $θ$ ) of a complex number, the real part is equal to $r \cos θ$ .
If you know the modulus ( $r$ ) and argument ( $θ$ ) of a complex number, how can you find the imaginary part?
If you know the modulus ( $r$ ) and argument ( $θ$ ) of a complex number, the imaginary part is equal to $r \sin θ$ .
What does a complex number look like when written in exponential (Euler) form?
A complex number that is written in exponential (Euler) form looks like $z = r e^{i θ}$ where $r$ is the modulus and $θ$ is the argument.
What is the complex conjugate of $r e^{i θ}$ ?
The complex conjugate of $r e^{i θ}$ is $r e^{- i θ}$ .
How can you write $z = r e^{i θ}$ in Cartesian form?
You can write $z = r e^{i θ}$ in Cartesian form as $z = r \cos θ + i r \sin θ$ .
True or False?
$e^{iπ} = - 1$ .
True.
$e^{iπ} = - 1$ .
True or False?
You can multiply and divide complex numbers written in exponential (Euler) form by using the laws of indices.
For example, $2 e^{\frac{π}{4} i} \times 3 e^{\frac{π}{2} i} = 6 e^{(\frac{π}{4} + \frac{π}{2}) i}$ .
True.
You can multiply and divide complex numbers written in exponential (Euler) form by using the laws of indices.
For example, $2 e^{\frac{π}{4} i} \times 3 e^{\frac{π}{2} i} = 6 e^{(\frac{π}{4} + \frac{π}{2}) i}$ .
How do you divide complex numbers written in modulus-argument (polar) form?
To divide complex numbers written in modulus-argument (polar) form, you divide the moduli and subtract the arguments.
$\frac{r_{1} (\cos (θ_{1}) + isin (θ_{1}))}{r_{2} (\cos (θ_{2}) + isin (θ_{2}))} = \frac{r_{1}}{r_{2}} (\cos (θ_{1} - θ_{2}) + isin (θ_{1} - θ_{2}))$ .
If $z = p + q i$ is a complex root to a polynomial with real coefficients, then what is another root of the polynomial?
If $z = p + q i$ is a complex root to a polynomial with real coefficients, then its complex conjugate, $z * = p - q i$ , is also a root.
True or False?
If a quadratic function has no real roots, then the real parts of the complex roots are equal to the $x$ -coordinate of the turning point of the graph.
True.
If a quadratic function has no real roots, then the real parts of the complex roots are equal to the $x$ -coordinate of the turning point of the graph.
If you are given a complex root ( $p + q i$ ) of a quadratic, how can you find the equation of the quadratic?
If you are given a complex root ( $p + q i$ ) of a quadratic, then you can find the equation of the quadratic by:
- writing as a product of two brackets and set it equal to zero $(x - (p + q i)) (x - (p - q i)) = 0$ ,
- rewriting the expressions to form a difference of two squares $((x - p) - q i) ((x - p) + q i) = 0$ ,
- expand and simplify ${(x - p)}^{2} + q^{2} = 0$ .
True or False?
A polynomial of degree $n$ has exactly $n$ distinct complex roots.
False.
A polynomial of degree $n$ has exactly $n$ complex roots, they are not necessarily distinct.
What is the smallest number of real roots that a cubic, with real coefficients, can have?
The smallest number of real roots that a cubic, with real coefficients, can have is one. It is not possible for a cubic to have zero real roots.
True or False?
A quartic, with real coefficients, can have at most two distinct non-real roots.
False.
A quartic, with real coefficients, can have at most four distinct non-real roots.
If you are given a complex root of a cubic with real coefficients, how do you find the real root?
If you are given a complex root of a cubic with real coefficients, you can find the real root by:
- find a quadratic which has the same complex root $(x - (p + q i)) (x - (p - q i))$ ,
- write the cubic as a product of the quadratic and a linear factor by comparing coefficients or using polynomial division,
- use the linear factor to find the real root.
What is de Moivre's Theorem?
De Moivre's Theorem tells us how to raise a complex number to a power.
If $z = r (\cos θ + isin θ)$ then $z^{n} = r^{n} (\cos (n θ) + isin (n θ))$
This can also be written as $z^{n} = r^{n} cis (n θ)$ or $z^{n} = r^{n} e^{i n θ}$ .
This is given in the formula booklet.
True or False?
The reciprocal of $z = \cos θ + isin θ$ is equal to its conjugate $z * = \cos θ - isin θ$ .
True.
The reciprocal of $z = \cos θ + isin θ$ is equal to its conjugate $z * = \cos θ - isin θ$ .
This is a result of de Moivre's Theorem: ${(\cos θ + isin θ)}^{- 1} = \cos (- θ) + isin (- θ)$ .
Proof by induction is used to prove de Moivre's Theorem, ${(\cos θ + isin θ)}^{n} = \cos (n θ) + isin (n θ)$ .
Part of the inductive step is $(\cos (k θ) + isin (k θ)) (\cos θ + isin θ)$ . How do you finish the inductive step?
Proof by induction is used to prove de Moivre's Theorem, ${(\cos θ + isin θ)}^{n} = \cos (n θ) + isin (n θ)$ .
Part of the inductive step is $(\cos (k θ) + isin (k θ)) (\cos θ + isin θ)$ . To finish the proof by induction, expand the brackets and use the compound angle identities to write $\cos (k θ) \cos θ - \sin (k θ) \sin θ = \cos ((k + 1) θ)$ and $\sin (k θ) \cos θ + \cos (k θ) \sin θ = \sin ((k + 1) θ)$ .
How can you find the square roots of a complex number without using de Moivre's Theorem?
For example, how could you find the square roots of $1 + 2 i$ ?
You can find the square roots of a complex number without using de Moivre's Theorem, by
- setting a square root equal to $c + d i$ ,
- squaring the expression and setting it equal to the original complex number ${(c + d i)}^{2} = a + b i$ ,
- form two equations using the real and imaginary parts and solve them.
For example, to find the square roots of $1 + 2 i$ , solve the equal ${(c + d i)}^{2} = 1 + 2 i$ .
How many cube roots does a complex number have?
A complex number has three cube roots.
What is an expression for the nth roots of the complex number $r e^{i θ}$ ?
The nth roots of the complex number $r e^{i θ}$ have the form $\sqrt[n]{r} e^{\frac{θ + 2 π k}{n} i}$ , where $k = 0, 1, 2, . . ., n - 1$ .
True or False?
If $ω$ is an nth root of a complex number then $ω e^{\frac{2 π}{n} i}$ is also an nth root.
True.
If $ω$ is an nth root of a complex number then $ω e^{\frac{2 π}{n} i}$ is also an nth root.
If you know one root you can find another by multiplying it by $e^{\frac{2 π}{n} i}$ .
True or False?
The nth roots of a complex number form a regular polygon when plotted on an Argand diagram.
True.
The nth roots of a complex number form a regular polygon when plotted on an Argand diagram.
True or False?
There is only one solution to the equation $z^{3} = 1 + 2 i$ .
False.
There are three solutions to the equation $z^{3} = 1 + 2 i$ . The solutions are the three cube roots of $1 + 2 i$ .

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