3.) •

2.)

1.) • f(x) = 0 when x = 3

• f(x) divided by (x + 1) has a remainder of 0

• f(−1) = 0

• (3x + 2) is a factor of f(x)

• (x - 3) is a factor of f(x)

Step-by-step explanation:

3.) By the Rational Root Theorem, we would take the Least Common Divisor [LCD] between the leading coefficient of 1, and the initial value of 5, which is 1, but we will take 5 it is a rational number we can work with; so this automatically makes our first factor of . Next, since the factor\divisor is in the form of , use what is called Synthetic Division. Remember, in this formula, −c gives you the OPPOSITE terms of what they really are, so do not forget it. Anyway, here is how it is done:

5| 1 −4 −6 4 5

↓ 5 5 −5 −5

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1 1 −1 −1 0 →

You start by placing the c in the top left corner, then list all the coefficients of your dividend [x⁴ - 4x³ - 6x² + 4x + 5]. You bring down the original term closest to c then begin your multiplication. Now depending on what symbol your result is tells you whether the next step is to subtract or add, then you continue this process starting with multiplication all the way up until you reach the end. Now, when the last term is 0, that means you have no remainder. Finally, your quotient is one degree less than your dividend, so that 1 in your quotient can be an x³, the x² follows right behind it, bringing −x right up against it, and bringing up the rear, −1, giving you the quotient of

However, we are not finished yet. This is our first quotient. The next step, while still using the Rational Root Theorem with our first quotient, is to take the Greatest Common Divisor [GCD] of the leading coefficient of 2, and the initial value of −1, which is 1, so this makes our next factor of . Then again, we use Synthetic Division because is in the form of :

1| 1 1 −1 −1

↓ 1 2 1

_________

1 2 1 0 →

So altogether, we have our four factors of

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2.) By the Rational Root Theorem again, this time, we will take 4,since the initial value is −8. This gives our automatic factor of Then start up Synthetic Division again:

4| 1 −1 −10 −8

↓ 4 12 8

___________

1 3 2 0 →

So altogether, we have our three factors of

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1.) By the Rational Root Theorem one more time, this time, we will take 3 since the initial value is −6. This gives our automatic factor of Then start up Synthetic Division again:

3| 3 −4 −13 −6

↓ 9 15 6

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3 5 2 0 →

Finally, you can just simply factor this second quotient:

So altogether, we have our three factors of and when set to equal zero, you will get With all the information given, you should be capable of figuring out the true statements.

I am joyous to assist you anytime.