Question

one root of (f(x) = 2x^3 + 9x^2 + 7x - 6) is (-3). explain how to find the factors of the polynomial.

Explanation:

🆕 New Concept Discovered: Factor Theorem and Polynomial Division
Using a known root to break down higher-degree polynomials.

Step 1: Identify the first factor from the given root

If \( x = c \) is a root of a polynomial, then by the Factor Theorem, \( (x - c) \) is a factor of that polynomial.

Since we are given that one root of \( f(x) = 2x^3 + 9x^2 + 7x - 6 \) is \( -3 \), the corresponding factor is:
\[ x - (-3) = x + 3 \]

Step 2: Divide the polynomial to find the remaining quadratic factor

To find the other factors, divide the polynomial \( f(x) \) by the known factor \( (x + 3) \) using either polynomial long division or synthetic division.

Using synthetic division with the root \( -3 \):

 -3 |   2    9    7   -6
    |       -6   -9    6
    --------------------
        2    3   -2    0

The quotient is the quadratic expression:
\[ 2x^2 + 3x - 2 \]

Step 3: Factor the remaining quadratic expression

Now, factor the quadratic quotient \( 2x^2 + 3x - 2 \). We look for two numbers that multiply to \( 2 \times (-2) = -4 \) and add up to \( 3 \). These numbers are \( 4 \) and \( -1 \).

Rewrite and factor by grouping:
\[ 2x^2 + 4x - x - 2 \]
\[ 2x(x + 2) - 1(x + 2) \]
\[ (2x - 1)(x + 2) \]

Step 4: Write the complete factored form

Combine all the factors found to write the fully factored polynomial:
\[ f(x) = (x + 3)(2x - 1)(x + 2) \]

Answer:

To find the factors of the polynomial:

1. Identify the first factor: Since \( -3 \) is a root, \( (x + 3) \) is a factor of the polynomial.
2. Divide the polynomial: Use synthetic division or long division to divide \( 2x^3 + 9x^2 + 7x - 6 \) by \( (x + 3) \). This yields the quadratic quotient \( 2x^2 + 3x - 2 \).
3. Factor the quadratic quotient: Factor \( 2x^2 + 3x - 2 \) into \( (2x - 1)(x + 2) \).
4. State the final factors: The complete factored form of the polynomial is \( (x + 3)(2x - 1)(x + 2) \).