Solving Cubic Polynomial Using Factor Theorem

Math Problem Statement

Solve the cubic polynomial 2x^3 + 3x^2 - 50x + 24 = 0, given that x = 4 is a root.

The problem provided involves solving a cubic polynomial using the Factor Theorem, with the root $x = 4$ . The polynomial is:

$2x^3 + 3x^2 - 50x + 24 = 0$

Identify the factor associated with the given root:
Since $x = 4$ is a root, by the Factor Theorem, $(x - 4)$ is a factor of the polynomial.
Use synthetic division to divide the polynomial by $(x - 4)$ . This will help factor the polynomial.

We will perform synthetic division using $x = 4$ for the polynomial $2x^3 + 3x^2 - 50x + 24$ .

Here’s how to do it:

4 & 2 & 3 & -50 & 24 \\ & & 8 & 44 & -24 \\ \hline & 2 & 11 & -6 & 0 \\ \end{array}$$ The quotient is $$2x^2 + 11x - 6$$ and the remainder is 0, confirming that $$(x - 4)$$ is a factor. 3. **Factor the quadratic**: Now we need to factor $$2x^2 + 11x - 6$$. This can be factored by looking for two numbers that multiply to $$2 \times (-6) = -12$$ and add to 11. These numbers are 12 and -1, so we can rewrite the middle term: $$2x^2 + 12x - x - 6$$ Now factor by grouping: $$2x(x + 6) - 1(x + 6)$$ Factor out $$(x + 6)$$: $$(2x - 1)(x + 6)$$ 4. **Final factorization**: The complete factorization of the polynomial is: $$(x - 4)(2x - 1)(x + 6) = 0$$ 5. **Solve for the roots**: Set each factor equal to zero: - $$x - 4 = 0 \Rightarrow x = 4$$ - $$2x - 1 = 0 \Rightarrow x = \frac{1}{2}$$ - $$x + 6 = 0 \Rightarrow x = -6$$ Thus, the roots of the polynomial are $$x = 4, \frac{1}{2}, -6$$. Would you like more details on any step? Here are 5 related questions: 1. What is synthetic division and how does it work? 2. How do you factor quadratic polynomials when the leading coefficient is not 1? 3. What other methods can be used to solve cubic polynomials? 4. What does the Factor Theorem tell us about the relationship between a polynomial and its roots? 5. How can you check your factorization by multiplying the factors back together? **Tip**: Always check your solution by substituting the roots back into the original equation to ensure they satisfy it.

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Cubic Polynomials
Factorization
Synthetic Division

Synthetic division
Factoring quadratic polynomials

Factor Theorem

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