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Featured 2 months ago

Using the

#color(blue)"law of logarithms"#

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(loga^n=nloga)color(white)(2/2)|)))#

#6^x=13# Take the ln ( natural log ) of both sides.

#rArrln6^x=ln13#

#rArrxln6=ln13#

#rArrx=ln13/ln6≈1.432" to 3 decimal places"#

Featured 2 months ago

#0# with multiplicity#3#

#2# with multiplicity#2#

#-2# with multiplicity#1#

#-1+sqrt(3)i# with multiplicity#1#

#-1-sqrt(3)i# with multiplicity#1#

The difference of squares identity can be written:

#a^2-b^2 = (a-b)(a+b)#

The difference of cubes identity can be written:

#a^3-b^3 = (a-b)(a^2+ab+b^2)#

We find:

#p(x) = (x^3-8)(x^5-4x^3)#

#color(white)(p(x)) = (x^3-2^3)x^3(x^2-2^2)#

#color(white)(p(x)) = (x-2)(x^2+2x+4)x^3(x-2)(x+2)#

#color(white)(p(x)) = x^3(x-2)^2(x+2)(x^2+2x+4)#

#color(white)(p(x)) = x^3(x-2)^2(x+2)(x^2+2x+1+3)#

#color(white)(p(x)) = x^3(x-2)^2(x+2)((x+1)^2-(sqrt(3)i)^2)#

#color(white)(p(x)) = x^3(x-2)^2(x+2)((x+1)-sqrt(3)i)((x+1)+sqrt(3)i)#

#color(white)(p(x)) = x^3(x-2)^2(x+2)(x+1-sqrt(3)i)(x+1+sqrt(3)i)#

Hence zeros:

#0# with multiplicity#3#

#2# with multiplicity#2#

#-2# with multiplicity#1#

#-1+sqrt(3)i# with multiplicity#1#

#-1-sqrt(3)i# with multiplicity#1#

Featured 2 months ago

The vertical asymptotes are

The slant asymptote is

No horozontal asymptote.

Let's factorise the denominator

The domain of

As we cannot divide by

So

The vertical asymptotes are

As the degree of the numerator is

Let's do a long division

So,

The slant asymptote is

To calculate the limits, we use the terms of highest degree.

There are no horizontal asymptote

When

When

graph{(y-(3x^3+1)/(4x^2-32))(y-x3/4)=0 [-28.86, 28.9, -14.43, 14.43]}

Featured 1 month ago

The answer is

If you want to simplify a quotient of complex numbers , multiply numerator and denominator by the conjugate of the denominator.

The conjugate of

And

Here,

So

Featured 4 weeks ago

We set up the long division of a polynomial by a simple monomial like this:

It works just like the long (numerical) division most of us learned back in elementary school, except now we're dividing with variables.

First we check: how many times does our leading

#color(white)(SPACE)3x^3#

#(x-2))bar(3x^4-4x^2+8x-1)#

Now, we multiply

#color(white)(SPACE)3x^3#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^3(x-2)#

#color(white)(SPACE)3x^3#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

We can subtract

Now, we check: how many times does our leading coefficient

#color(white)(SPACE)3x^3+6x^2#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACE)-6x^2(x-2)#

#color(white)(SPACE)3x^3+6x^2#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACES)-6x^3+12x^2#

This gets rid of our

#color(white)(SPACE)3x^3+6x^2#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPA)+12x^2#

#color(white)(SPACE)3x^3+6x^2#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPAC)8x^2#

Next we check: how many times does our leading coefficient

#color(white)(SPACE)3x^3+6x^2+8x#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPAC)8x^2#

#color(white)(SPACESP)-8x(x-2)#

#color(white)(SPACE)3x^3+6x^2+8x#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPAC)8x^2#

#color(white)(SPACESP)-8x^2+16x#

#color(white)(SPACE)3x^3+6x^2+8x#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPAC)8x^2#

#color(white)(SPACESPACES)+16x#

Now we add

#color(white)(SPACE)3x^3+6x^2+8x#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPAC)8x^2#

#color(white)(SPACESPACESPA)24x#

Next we check: how many times does our leading coefficient

#color(white)(SPACE)3x^3+6x^2+8x+24#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPAC)8x^2#

#color(white)(SPACESPACESPA)24x#

#color(white)(SPACESPACE)-24(x-2)#

#color(white)(SPACE)3x^3+6x^2+8x+24#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPAC)8x^2#

#color(white)(SPACESPACESPA)24x#

#color(white)(SPACESPACE)-24x+48#

#color(white)(SPACE)3x^3+6x^2+8x+24#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPAC)8x^2#

#color(white)(SPACESPACESPA)24x#

#color(white)(SPACESPACESPACE)48#

Now we add

#color(white)(SPACE)3x^3+6x^2+8x+24#

#(x-2))bar(3x^4-4x^2+8x-1)#

#color(white)(SPA)-3x^4+6x^3#

#color(white)(SPACESPA)6x^3#

#color(white)(SPACESPAC)8x^2#

#color(white)(SPACESPACESPA)24x#

#color(white)(SPACESPACESPACE)47#

And of course,

The final answer is therefore

Featured 3 weeks ago

A reflection about the

For any function

When we take that output

Let's say that when

Thus, when

All that happens is the sign of the output value changes—points that were once above the

This is simply stated as a reflection where the mirror is the

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