Find the real zeros of f(x); there are 4 zeros

x = 4 or x = 1 or x = -2 or x = -3/2

Step-by-step explanation:

Solve for x over the real numbers:

2 x^4 - 3 x^3 - 21 x^2 - 2 x + 24 = 0

The left hand side factors into a product with four terms:

(x - 4) (x - 1) (x + 2) (2 x + 3) = 0

Split into four equations:

x - 4 = 0 or x - 1 = 0 or x + 2 = 0 or 2 x + 3 = 0

Add 4 to both sides:

x = 4 or x - 1 = 0 or x + 2 = 0 or 2 x + 3 = 0

Add 1 to both sides:

x = 4 or x = 1 or x + 2 = 0 or 2 x + 3 = 0

Subtract 2 from both sides:

x = 4 or x = 1 or x = -2 or 2 x + 3 = 0

Subtract 3 from both sides:

x = 4 or x = 1 or x = -2 or 2 x = -3

Divide both sides by 2:

 x = 4 or x = 1 or x = -2 or x = -3/2

Step-by-step explanation:

By the Rational Root Theorem, we take the Least Common Divisor [LCD] between the leading coefficient of 2, and the initial value of 24, which is 1, 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:

1| 2 −3 −21 −2 24

↓ 2 −1 −22 −24

_______________

2 −1 −22 −24 0 →

You start by placing the c in the top left corner, then list all the coefficients of your dividend [2x⁴ - 3x³ - 21x² - 2x + 24]. 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 2 in your quotient can be a 2x³, the −x² follows right behind it, bringing −22x right up against it, and bringing up the rear, −24, 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 −24, which in this case would be −2 because the negative overpowers everything, so this makes our next factor of . Then again, we use Synthetic Division because is in the form of :

−2| 2 −1 −22 −24

↓ −4 10 24

____________

2 −5 −12 0 →

Finally, you can just simply factor this second quotient:

So altogether, we have our four factors of , then setting all factors equal to zero, you get all the values of .

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step-by-step explanation:

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