https://file.aiquickdraw.com/mathbot/file/1729573370230q68l33ky.jpg

We are given the polynomial:

\[
g(x) = x^3 - x^2 + 17x + 87
\]

and the fact that \(-3\) is a zero of the polynomial. We need to express \( g(x) \) as a product of linear factors.

### Step 1: Use synthetic division to divide \( g(x) \) by \( (x + 3) \).

Since \(-3\) is a zero of the polynomial, \( (x + 3) \) is a factor of \( g(x) \).

Let's set up synthetic division with \( -3 \) as the divisor:

The coefficients of \( g(x) = x^3 - x^2 + 17x + 87 \) are \( [1, -1, 17, 87] \).

Performing synthetic division:

1. Write down the coefficients: \( 1, -1, 17, 87 \).
2. Bring down the first coefficient \( 1 \).
3. Multiply \( 1 \times -3 = -3 \), then add to the next coefficient: \( -1 + (-3) = -4 \).
4. Multiply \( -4 \times -3 = 12 \), then add to the next coefficient: \( 17 + 12 = 29 \).
5. Multiply \( 29 \times -3 = -87 \), then add to the next coefficient: \( 87 + (-87) = 0 \).

The synthetic division table looks like this:

\[
\begin{array}{r|rrrr}
-3 & 1 & -1 & 17 & 87 \\
   &   & -3 & 12 & -87 \\
\hline
   & 1 & -4 & 29 & 0 \\
\end{array}
\]

The quotient is \( x^2 - 4x + 29 \), and the remainder is \( 0 \), confirming that \( x + 3 \) is a factor.

### Step 2: Factor \( x^2 - 4x + 29 \).

Now we need to check if \( x^2 - 4x + 29 \) can be factored further. Since the discriminant of this quadratic is:

\[
\Delta = (-4)^2 - 4(1)(29) = 16 - 116 = -100
\]

which is negative, the quadratic does not factor over the real numbers. However, it does have complex roots.

The roots of \( x^2 - 4x + 29 \) are:

\[
x = \frac{-(-4) \pm \sqrt{-100}}{2(1)} = \frac{4 \pm 10i}{2} = 2 \pm 5i
\]

Thus, the quadratic factors as \( (x - (2 + 5i))(x - (2 - 5i)) \).

### Step 3: Express \( g(x) \) as a product of linear factors.

We now have the factorization of \( g(x) \):

\[
g(x) = (x + 3)(x - (2 + 5i))(x - (2 - 5i))
\]

This is the factorization of \( g(x) \) as a product of linear factors over the complex numbers.

Would you like further details or explanations?

Here are five related questions:
1. How do you perform synthetic division in more detail?
2. What is the importance of the discriminant in factoring quadratics?
3. How are complex roots derived using the quadratic formula?
4. Why do we express polynomials in terms of linear factors?
5. Can this polynomial be factored over real numbers only?

**Tip**: Always check the discriminant when trying to factor a quadratic equation. It helps determine whether the roots are real or complex.

Express g(x) = x^3 - x^2 + 17x + 87 as a product of linear factors, given that -3 is a zero of the polynomial.

Learn how to factor the cubic polynomial g(x) = x^3 - x^2 + 17x + 87 using synthetic division. We show the step-by-step process, including finding the quadratic form and expressing the polynomial in terms of complex linear factors. This problem is suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation