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Binomial Theorem will follow this pattern:
The powers start at 3 and decrease down to 0 for the first term,
and start at 0 and increase to 3 for the other term.
(2x+4)^3 =
See how the power counts down on the (2x)
and counts up on the 4?
That's the pattern that our expansion must follow.
I left a little space in front of each term.
The coefficient in front of each term will come from the fourth row of Pascal's Triangle. The one that looks like this:
1 3 3 1
Those are the coefficients we want:
We can clean this up a little bit by getting rid of some of the junk. Anything to the 0th power is 1. So let's suppress all of our 1's because multiplying by 1 is not important.
Now apply exponent rule, distributing the power to both the 2 and the x where applicable.
Remember, multiplication is COMMUTATIVE, meaning we can multiply things in any order. So let's bring the numerical portion to the front of each term and multiply it all out.
Binomial Theorem will follow this pattern:
The powers start at 3 and decrease down to 0 for the first term,
and start at 0 and increase to 3 for the other term.
(2x+4)^3 =
See how the power counts down on the (2x)
and counts up on the 4?
That's the pattern that our expansion must follow.
I left a little space in front of each term.
The coefficient in front of each term will come from the fourth row of Pascal's Triangle. The one that looks like this:
1 3 3 1
Those are the coefficients we want:
We can clean this up a little bit by getting rid of some of the junk. Anything to the 0th power is 1. So let's suppress all of our 1's because multiplying by 1 is not important.
Now apply exponent rule, distributing the power to both the 2 and the x where applicable.
Remember, multiplication is COMMUTATIVE, meaning we can multiply things in any order. So let's bring the numerical portion to the front of each term and multiply it all out.
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The answer is, 
I'm about to basically type up a report on explaining this answer so bear with me :P
<em>Step 1.)</em> Factor
using Polynomial Division.
Just a basic rundown of what polynomial division is. When using polynomial division for this problem we find all the factors of our constant, 84, and then see which one, when substituted for x, makes the expression equal zero. After we find our variable that makes the problem equal zero, we divide it into our original expression. Here's how it's done:
These are our factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
strating from one and plugging in these numbers, the first one we come upon that works is 4. So, now we will divide. below is a picture of what it should look like.
After we divide, we get an answer of
<em>Step 2.)</em> split into two terms
To do that we multiply the coefficient of the first term (2) by the constant (-21)
So:
Now we ask "which two numbers add up to equal 11, but multiply to equal -42". Those two numbers are 14 and -3.
Now we basically split 11x as the sum of 14 and -3. So now our problem should look like this:
<span><span>
</span></span><em>Step 3.)</em> Factor out the common terms in the first two terms, then the last two like this:
From this:
To this:
<em>Step 4.)</em> Factor the term (x + 7) out of the parenthesis
After we do that, we are left with our problem looking like this:
The last thing we do is set each parenthesis equal to zero and work it out to find our zeros.
That gives us our answer of our zeros being
Thank you for our question! I hope this helped! Have an amazing day! :D
I'm about to basically type up a report on explaining this answer so bear with me :P
<em>Step 1.)</em> Factor
Just a basic rundown of what polynomial division is. When using polynomial division for this problem we find all the factors of our constant, 84, and then see which one, when substituted for x, makes the expression equal zero. After we find our variable that makes the problem equal zero, we divide it into our original expression. Here's how it's done:
These are our factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
strating from one and plugging in these numbers, the first one we come upon that works is 4. So, now we will divide. below is a picture of what it should look like.
After we divide, we get an answer of
<em>Step 2.)</em> split
To do that we multiply the coefficient of the first term (2) by the constant (-21)
So:
Now we ask "which two numbers add up to equal 11, but multiply to equal -42". Those two numbers are 14 and -3.
Now we basically split 11x as the sum of 14 and -3. So now our problem should look like this:
<span><span>
</span></span><em>Step 3.)</em> Factor out the common terms in the first two terms, then the last two like this:
From this:
To this:
<em>Step 4.)</em> Factor the term (x + 7) out of the parenthesis
After we do that, we are left with our problem looking like this:
The last thing we do is set each parenthesis equal to zero and work it out to find our zeros.
That gives us our answer of our zeros being
Thank you for our question! I hope this helped! Have an amazing day! :D
The distance from the sun is option 2 5.59 astronomical units.
Step-by-step explanation:
Step 1; To solve the question we need two variables. P which represents the number of years a planet takes to complete a revolution around the Sun. This is given as 13.2 years in the question so P = 13.2 years. The other variable is the distance between the planet and the sun in astronomical units. We need to determine the value of this using the given equation.
Step 2; So we have to calculate the value of 'a' in Kepler's equation. But the exponential power is on the variable we need to find so we multiply both the sides by an exponential power of
to be able to calculate 'a'.
P = ,
=
,
= a,
= a = 5.58533 astronomical units.
Rounding it over to nearest hundredth we get 5.59 astronomical units.