- Identify that \((x-3)\) is a factor of the cubic polynomial \(x^3 - 4x^2 - 3x + 18 = 0\) since \(x=3\) is a root.
- Use polynomial division to find the quotient \((x^2 - x - 6)\).
- Factor the quadratic quotient \((x^2 - x - 6)\) into \((x-3)(x+2)\).
- Combine all factors to get \((x+2)(x-3)^2=0\), which matches the second option.
- The final answer is \(\boxed{(x+2)(x-3)^2=0}\).
Explanation
- Problem Analysis
We are given the cubic polynomial equation \(x^3 - 4x^2 - 3x + 18 = 0\). We are also given that \(x = 3\) is a zero of this equation. According to the Factor Theorem, if \(x = c\) is a zero of a polynomial \(P(x)\), then \((x - c)\) is a factor of \(P(x)\). Thus, \((x - 3)\) must be a factor of the given polynomial. Our goal is to determine the complete factorization of the cubic expression by dividing it by its known factor \((x - 3)\) and then factoring the resulting quadratic.
- Polynomial Division
To find the other factors, we divide the cubic polynomial \(P(x) = x^3 - 4x^2 - 3x + 18\) by \((x - 3)\). Performing polynomial division:
Dividing \(x^3\) by \(x\) gives \(x^2\). Multiplying \(x^2(x - 3) = x^3 - 3x^2\).
Subtracting \((x^3 - 4x^2 - 3x + 18) - (x^3 - 3x^2) = -x^2 - 3x + 18\).
Dividing \(-x^2\) by \(x\) gives \(-x\). Multiplying \(-x(x - 3) = -x^2 + 3x\).
Subtracting \((-x^2 - 3x + 18) - (-x^2 + 3x) = -6x + 18\).
Dividing \(-6x\) by \(x\) gives \(-6\). Multiplying \(-6(x - 3) = -6x + 18\).
Subtracting gives a remainder of \(0\).
The quotient obtained is \(x^2 - x - 6\). Thus, the polynomial can be written as \((x - 3)(x^2 - x - 6) = 0\).
- Factoring the Quadratic Quotient
Now we factor the resulting quadratic expression \(x^2 - x - 6\). We look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\). Therefore, we can write the quadratic as:
\[x^2 - x - 6 = (x - 3)(x + 2)\]
Substituting this back into our expression, the full factored form is:
\[(x - 3)(x - 3)(x + 2) = (x + 2)(x - 3)^2 = 0\]
- Conclusion
Comparing our result \((x + 2)(x - 3)^2 = 0\) with the given options:
- \((x+2)(x-\sqrt{3})(x+\sqrt{3})=0\)
- \((x+2)(x-3)^2=0\)
- \((x-2)(x-3)^2=0\)
- \((x-2)(x-3)(x+3)=0\)
The correct option is the second one. Thus, the factored form is \((x+2)(x-3)^2=0\).
Examples
Factoring polynomials is a fundamental skill in algebra with applications in fields like engineering and economics. For example, in structural engineering, determining the roots of a polynomial can help find the points where a beam experiences zero bending moment, which is critical for safety. By finding the factored form, we can easily identify these points and design stronger, more efficient structures. Similarly, in business, finding the roots of a profit equation can help identify the "break-even" points where revenue equals costs.