## Binomial Product

In polar form, what is the product of the following two binomials?

##
__
__**Hint**

**Hint**

Use the FOIL (

**F**irst,**O**uter,**I**nner,**L**ast) method to multiply the two binomials (an expression with two terms). Remember to track +/- signs.$$$(a+b)(c+d)=ac+ad+bc+bd$$$

##
__
__**Hint 2**

**Hint 2**

Polar form is:

$$$z=c\: \angle \: \theta $$$

where
$$c=\sqrt{a^2+b^2}$$
(the vector’s length) and
$$\theta=tan^{-1}(b/a)$$
. Note
$$a$$
is the real component and
$$b$$
is the imaginary component of the rectangular form.

Use the FOIL (

**F**irst,**O**uter,**I**nner,**L**ast) method to multiply the two binomials (an expression with two terms). Remember to track +/- signs.$$$(a+b)(c+d)=ac+ad+bc+bd$$$

Thus,

$$$(2-5i)(\sqrt{3}+8i)=2\sqrt{3}+16i-5\sqrt{3}i-40i^2$$$

Remember
$$i=\sqrt{-1}$$
, so we can simplify:

$$$=3.46+16i-8.66i-40(-1)$$$

$$$=3.46+40+7.34i=43.46+7.34i$$$

Complex numbers can be written in both rectangular and polar form. For rectangular form, a complex number is represented by both its real and imaginary components:

$$$z=a+jb$$$

where
$$a$$
is the real component,
$$b$$
is the imaginary component, and
$$j=\sqrt{-1}$$
which is often expressed as
$$i=\sqrt{-1}$$
.

Analyzing our rectangular coordinate form
$$43.46+7.34i$$
, we can say
$$a=43.46$$
and
$$b=7.34$$
.

Polar form is:

$$$z=c\: \angle \: \theta $$$

where
$$c=\sqrt{a^2+b^2}$$
(the vector’s length) and
$$\theta=tan^{-1}(b/a)$$
. Note:
$$a=c\:cos(\theta)$$
and
$$b=c\:sin(\theta)$$
.

Therefore,

$$$c=\sqrt{43.46^2+7.34^2}=\sqrt{1,888.77+53.876}=\sqrt{1,942.65}=44.08$$$

$$$\theta=tan^{-1}(\frac{7.34}{43.46})=9.6^{\circ}$$$

Finally, the polar form is:

$$$44.08 \: \angle \: 9.6^{\circ}$$$

$$$44.08 \: \angle \: 9.6^{\circ}$$$