We are tasked with finding all the roots of the polynomial $f(x) = x^3 - 9x^2 + 26x - 24$, given that $x = 2$ is one of its roots. Let's solve this using the Remainder Theorem and synthetic division.

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Step 1: Verify that $x = 2$ is a root

The Remainder Theorem states that if $x = 2$ is a root, substituting $x = 2$ into $f(x)$ will yield $f(2) = 0$. Let us verify this:

$f(2) = 2^3 - 9(2^2) + 26(2) - 24$ $= 8 - 36 + 52 - 24 = 0$

Since $f(2) = 0$, $x = 2$ is indeed a root.

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Step 2: Perform synthetic division by $x - 2$

Divide $f(x)$ by $x - 2$ using synthetic division:

Polynomial: $f(x) = x^3 - 9x^2 + 26x - 24$

• Coefficients: $[1, -9, 26, -24]$
• Divisor root: $x = 2$

Perform the division:

2 & 1 & -9 & 26 & -24 \\ & & 2 & -14 & 24 \\ \hline & 1 & -7 & 12 & 0 \\ \end{array}$$ The quotient is $$x^2 - 7x + 12$$, and the remainder is $$0$$, confirming that $$x - 2$$ divides $$f(x)$$ completely. --- ### Step 3: Factor the quotient $$x^2 - 7x + 12$$ The quadratic $$x^2 - 7x + 12$$ can be factored easily: $$x^2 - 7x + 12 = (x - 3)(x - 4)$$ --- ### Step 4: Combine all roots The complete factorization of $$f(x)$$ is: $$f(x) = (x - 2)(x - 3)(x - 4)$$ Thus, the roots of $$f(x)$$ are: $$x = 2, \, x = 3, \, x = 4$$ --- ### Final Answer: The correct choice is: $$\boxed{x = 2, \, x = 3, \, x = 4}$$ --- Would you like a detailed explanation of synthetic division or factoring? ### Related Questions: 1. How does the Remainder Theorem help verify roots of polynomials? 2. Can all cubic equations be solved using synthetic division? 3. How do you determine if a quadratic factor has real roots? 4. Why is $$x = 2$$ called a "root" of the function $$f(x)$$? 5. How can we use the Rational Root Theorem to find roots systematically? ### Tip: For polynomials of degree 3 or higher, always try to simplify using known roots and synthetic division to reduce the degree!