Question

Find a cubic function with the given zeros.

Square root of two., - Square root of two., -2

f(x) = x3 + 2x2 - 2x + 4
f(x) = x3 + 2x2 + 2x - 4
f(x) = x3 - 2x2 - 2x - 4
f(x) = x3 + 2x2 - 2x - 4
@amistre64

Answer 1

Please help! @robtobey

Answer 2

Refer to the attachment.

Answer 3

D.

Answer 4

Find the zeros of the polynomial function and state the multiplicity of each.

f(x) = 5(x + 7)2(x - 7)3

-7, multiplicity 2; 7, multiplicity 3
4, multiplicity 1; -7, multiplicity 3; 7, multiplicity 3
-7, multiplicity 3; 7, multiplicity 2
4, multiplicity 1; 7, multiplicity 1; -7, multiplicity 1

Answer 5

@iGreen pleaase help!!

Answer 6

For the first question above, a polynomial function with roots x1, x2, x3, ...
has equation
(x - x1)(x - x2)(x - x3)... = 0
Write all the binomial factors using the roots you were given, then multiply them out.

Answer 7

For the second question, from the choices it is obvious some of those numbers are exponents.
Is the equation
\(f(x) = 5(x + 7)^2(x - 7)^3\)
If so, a factor of x + 7 will give a root of x = -7 since
x + 7 = 0
x = -7
A factor of x - 7 give x = 7 since
x - 7 = 0
x = 7
The multiplicity comes from the exponent in each factor.
Since the factor x + 7 has an exponent of 2, x = -7 has multiplicity 2.
Since the factor x - 7 has exponent 3, x = 7 has multiplicity 3.

Answer 8

Think of it this way:

\(f(x) = 5(x + 7)(x + 7)(x - 7)(x - 7)(x - 7)\)

Set the function equal to zero:

\(5(x + 7)(x + 7)(x - 7)(x - 7)(x - 7) = 0\)

Now set each factor that has x in it equal to zero:

\(x + 7 = 0\) or \(x + 7 = 0\) or \(x - 7 = 0\) or \(x - 7 = 0\) or \(x - 7 = 0\)

Solve all the equations:

x = -7 or x = -7 or x = 7 or x = 7 or x = 7 or x = 7

Now count how many roots are -7 and how many are 7:
2 roots -7
3 roots 7
Answer is:
-7, multiplicity 2; 7, multiplicity 3

Answer 9

Thank you so much! @mathstudent55

Answer 10

You're welcome.