- Given that \(-3\) is a root of \(f(x) = 2x^3 + 9x^2 + 7x - 6\), divide the polynomial by \((x + 3)\) to find the quotient \(2x^2 + 3x - 2\).
- Factor the quadratic quotient \(2x^2 + 3x - 2\) into \((2x - 1)(x + 2)\).
- Express the fully factored form as \(f(x) = (x + 3)(2x - 1)(x + 2)\).
- The three correct factors from the provided list are \(\boxed{2x - 1, x + 3, x + 2}\).
Explanation
- Analyzing the Polynomial and Dividing by the Known Factor
The polynomial provided is \(f(x) = 2x^3 + 9x^2 + 7x - 6\). We are given that \(x = -3\) is a root, which implies, by the Factor Theorem, that \((x - (-3)) = (x + 3)\) is a factor of the polynomial. To find the other factors, we can perform polynomial division by dividing \(f(x)\) by \((x + 3)\).
- Performing Polynomial Division
Performing synthetic division of \(f(x) = 2x^3 + 9x^2 + 7x - 6\) by \((x + 3)\):
Coefficients of \(f(x)\) are \(2, 9, 7, -6\).
Setting up the synthetic division with the root \(-3\):
- Bring down the 2.
- \(2 \times (-3) = -6\). Adding to 9 gives \(9 + (-6) = 3\).
- \(3 \times (-3) = -9\). Adding to 7 gives \(7 + (-9) = -2\).
- \(-2 \times (-3) = 6\). Adding to -6 gives \(-6 + 6 = 0\).
The quotient is a quadratic polynomial \(2x^2 + 3x - 2\). Thus, \(f(x) = (x + 3)(2x^2 + 3x - 2)\).
- Factoring the Quadratic Quotient
Now we factor the quadratic quotient \(2x^2 + 3x - 2\). We look for two numbers that multiply to \(2 \times (-2) = -4\) and add to \(3\). Those numbers are \(4\) and \(-1\).
Rewriting the middle term: \(2x^2 + 4x - x - 2\).
Factor by grouping: \(2x(x + 2) - 1(x + 2) = (2x - 1)(x + 2)\).
Thus, the fully factored form of the polynomial is \(f(x) = (x + 3)(2x - 1)(x + 2)\).
- Identifying Correct Factors and Conclusion
From the expression \(f(x) = (x + 3)(2x - 1)(x + 2)\), the three linear factors are \((x + 3)\), \((2x - 1)\), and \((x + 2)\). Comparing these with the provided options:
- \(2x - 1\) is correct.
- \(x + 3\) is correct.
- \(x + 2\) is correct.
Note that \(-2x + 1\) is also technically a factor because \(-(2x - 1) = -2x + 1\), but we have identified our three chosen factors. Therefore, the three correct choices from the list are \((2x - 1)\), \((x + 3)\), and \((x + 2)\).
Examples
Understanding polynomial factorization is essential in various fields such as engineering, physics, and economics. For example, if you are modeling the trajectory of a projectile, the factors of the polynomial representing its height can tell you when it hits the ground (the roots). Similarly, in electrical engineering, factoring polynomials helps in analyzing circuit stability. Mastering these skills allows you to decompose complex systems into simpler components that are easier to analyze and solve.