Given P(x)=3x5−2x4+96x3−64x2−432x+288P(x)=3x5-2x4+96x3-64x2-432x+288, and that 6i6i is a zero, write PP in factored form (as a product of linear factors). Be sure to write the full equation, including P(x)=P(x)=.

Given the polynomial \( P(x) = 3x^5 - 2x^4 + 96x^3 - 64x^2 - 432x + 288 \) and the information that \( 6i \) is a zero, we can proceed as follows.

### Step 1: Identify the Conjugate Pair
Since \( 6i \) is a zero of the polynomial and the coefficients of the polynomial are real, its conjugate \( -6i \) must also be a zero of the polynomial.

### Step 2: Form the Quadratic Factor
The zeros \( 6i \) and \( -6i \) give rise to a quadratic factor. The quadratic factor corresponding to these imaginary roots is:

\[
(x - 6i)(x + 6i) = x^2 + 36
\]

### Step 3: Perform Polynomial Division
To find the remaining factors of \( P(x) \), we need to divide \( P(x) \) by the quadratic factor \( x^2 + 36 \). This division will give a quotient that can be factored further if possible.

Let's divide \( P(x) \) by \( x^2 + 36 \).

#### Performing the division:

We are dividing:
\[
P(x) = 3x^5 - 2x^4 + 96x^3 - 64x^2 - 432x + 288
\]
by
\[
x^2 + 36
\]

#### Quotient:
The result of the division is \( 3x^3 - 2x^2 + 24x - 8 \).

### Step 4: Factor the Quotient
Now, we need to factor \( 3x^3 - 2x^2 + 24x - 8 \). We can apply synthetic division or factor by grouping. The cubic can be factored as:

\[
3x^3 - 2x^2 + 24x - 8 = (x - 2)(3x^2 + 4x + 4)
\]

### Step 5: Factor the Remaining Quadratic
Now we check whether the quadratic \( 3x^2 + 4x + 4 \) can be factored further. Since its discriminant \( \Delta = 4^2 - 4(3)(4) = 16 - 48 = -32 \) is negative, the quadratic cannot be factored further over the real numbers. Therefore, it remains as it is.

### Final Factored Form
Thus, the full factorization of \( P(x) \) is:

\[
P(x) = 3(x^2 + 36)(x - 2)(3x^2 + 4x + 4)
\]

This is the factored form of the polynomial \( P(x) \).

Would you like further clarification or more detailed steps?

---

Here are five questions that could expand on this information:
1. How do we use synthetic division for polynomials?
2. What is the significance of complex conjugate pairs in polynomials?
3. How can you determine the discriminant of a quadratic?
4. Why are some quadratics not factorable over the real numbers?
5. How does polynomial long division work for dividing large polynomials?

**Tip**: Always check for the discriminant of a quadratic when factoring to determine whether it has real or complex roots.

Given P(x)=3x^5−2x^4+96x^3−64x^2−432x+288, and that 6i is a zero, write P(x) in factored form (as a product of linear factors). Be sure to write the full equation, including P(x)=

Explore the full factorization of the polynomial P(x)=3x^5−2x^4+96x^3−64x^2−432x+288, given that 6i is a zero. This problem involves complex roots, conjugate pairs, polynomial division, and cubic factoring. Ideal for students in Grades 11-12, with detailed step-by-step explanations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation