Math Problem Statement

A polynomial function f(x) with real coefficients has the given degree, zeros, and solution point.

Degree

Zeros

Solution Point

4

−1, 5, i

f(1) = 48

(a) Write the function in completely factored form.

f(x) =

Solution

To find the polynomial function f(x)f(x) with the given information, we proceed as follows:

Given:

  • Degree: 4
  • Zeros: 1-1, 55, and ii
  • Solution point: f(1)=48f(1) = 48

Since the polynomial has real coefficients, the complex conjugate i-i must also be a zero. The zeros are 1-1, 55, ii, and i-i.

Step 1: Form the factors

The zeros give us the factors of the polynomial: (x+1),(x5),(xi),(x+i)(x + 1), \quad (x - 5), \quad (x - i), \quad (x + i)

Step 2: Write the function in factored form

The polynomial can be written as: f(x)=a(x+1)(x5)(xi)(x+i)f(x) = a (x + 1)(x - 5)(x - i)(x + i) where aa is a constant to be determined.

Step 3: Simplify the complex conjugate factors

Notice that: (xi)(x+i)=x2+1(x - i)(x + i) = x^2 + 1 Thus, the polynomial becomes: f(x)=a(x+1)(x5)(x2+1)f(x) = a (x + 1)(x - 5)(x^2 + 1)

Step 4: Determine the value of aa

Use the given solution point f(1)=48f(1) = 48: f(1)=a(1+1)(15)(12+1)=48f(1) = a (1 + 1)(1 - 5)(1^2 + 1) = 48 a(2)(4)(2)=48a (2)(-4)(2) = 48 16a=48-16a = 48 a=3a = -3

Step 5: Write the final function

The polynomial in completely factored form is: f(x)=3(x+1)(x5)(x2+1)f(x) = -3 (x + 1)(x - 5)(x^2 + 1)

Would you like details on how to expand this polynomial, or have any other questions?

Related Questions:

  1. How would you expand f(x)=3(x+1)(x5)(x2+1)f(x) = -3(x + 1)(x - 5)(x^2 + 1) into standard form?
  2. What happens to the zeros if the polynomial's coefficients are not real?
  3. How do complex conjugate pairs affect the roots of a polynomial?
  4. What would be the steps to find the derivative of f(x)f(x)?
  5. How can you verify that f(x)=3(x+1)(x5)(x2+1)f(x) = -3(x + 1)(x - 5)(x^2 + 1) satisfies f(1)=48f(1) = 48?

Tip:

When given zeros of a polynomial with real coefficients, always include the conjugate pair for any complex zeros.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Complex Numbers
Real Coefficients
Factoring

Formulas

f(x) = a(x + 1)(x - 5)(x - i)(x + i)
(x - i)(x + i) = x^2 + 1

Theorems

Fundamental Theorem of Algebra
Complex Conjugate Root Theorem

Suitable Grade Level

Grades 10-12