Question

what are the zeros of the polynomial function? f(x) = x⁴ − 4x³ − 22x² 4x 21 select each correct answer. responses a. -3 b. -1 c. 0 d. 1 e. 3 f. 7

Answer 1

The zeros of the polynomial function f(x) = x⁴ - 4x³ - 22x² + 4x + 21 are x = -3 and x = 7. These values are found by testing potential zeros and verifying which ones satisfy the equation f(x) = 0.

Step-by-step explanation:

To find the zeros of the polynomial function f(x) = x⁴ − 4x³ − 22x² + 4x + 21, we need to find the values of x that make f(x)=0. If we can factor the polynomial, that's typically the easiest method. Otherwise, we can try synthetic division or apply the Rational Root Theorem to find potential rational zeros and test them.

If we notice that when we plug in the potential zeros, such as x = -3, x = -1, x = 1, and x = 3, into the function, we get:

f(-3) = (-3)⁴ - 4(-3)³ - 22(-3)² + 4(-3) + 21 = 81 + 108 - 198 - 12 + 21 = 0

f(-1) = (-1)⁴ - 4(-1)³ - 22(-1)² + 4(-1) + 21 != 0

f(1) = (1)⁴ - 4(1)³ - 22(1)² + 4(1) + 21 != 0

f(3) = (3)⁴ - 4(3)³ - 22(3)² + 4(3) + 21 = 81 - 108 - 198 + 12 + 21 != 0

We find that only x = -3 and x = 7 are zeros of the function since they result in f(x)=0. Therefore, the correct answers are responses A and F.

Answer 2

The zeros of the polynomial of the given function is x=7,1,−1,−3. The correct responses are a,b,d,f.

Let's find the zeros of the polynomial function:

`f(x)=x^4-4 x^3-22 x^2+4 x+21`

We can solve the equation by factoring the polynomial and setting each factor equal to zero.

Factoring the polynomial and equating to zero, we get the equation as follows:

`(x-7)\left(x^3+3 x^2-x-3\right)=0`

Factor the remaining polynomial by further simplifying:

`(x-7)(x-1)\left(x^2+4 x+3\right)=0`

Factoring the quadratic part and combining this with other factored parts, we get:

`(x-7)(x-1)(x+1)(x+3)=0`

Set each factor equal to zero and then solve:

x−7=0⇒x=7

x−1=0⇒x=1

x+1=0⇒x=−1

x+3=0⇒x=−3

Finally, we got the required value of x i.e., zeros of the polynomial to be:

x=7,1,−1,−3

Question:

what are the zeros of the polynomial function?

f(x) = x⁴ − 4x³ − 22x² + 4x + 21. Select each correct answer from the following responses.

a. -3

b. -1

c. 0

d. 1

e. 3

f. 7