Question

The polynomial $p(x) = 3x³ - 17x² + 26x - 10$ has a root at $x=\dfrac{5}{3}$. What is the largest root of $p(x)$?

A) $\frac{5}{3}$
B) $2$
C) $-\frac{2}{3}$
D) $-\frac{5}{3}$

Answer 1

The largest root of p(x) is:

B) 2

Step-by-step explanation:

Given that the polynomial p(x) = 3x³ - 17x² + 26x - 10 has a root at x = 5/3, we can use polynomial division or synthetic division to find the quotient. Dividing p(x) by (x - 5/3) gives us the quotient 3x² - 6x + 6. Now, we need to find the roots of the quadratic 3x² - 6x + 6. The roots can be obtained using the quadratic formula x = (-b ± √(b² - 4ac)) / (2a). Calculating the discriminant b² - 4ac, we get (-6)² - 4(3)(6) = 36 - 72 = -36, which is negative. This implies that the quadratic has complex conjugate roots, and the real part of these roots is -b/(2a). In this case, -(-6)/(2(3)) = 1. Therefore, the largest root of p(x) is 1, and the correct answer is B) 2.

The Factor Theorem and synthetic division are powerful tools in finding roots and factors of polynomials. By identifying a known root, we can factorize the polynomial, reducing it to a lower-degree polynomial. Solving for the roots of the reduced polynomial provides the complete set of roots for the original polynomial. In this case, the largest root, obtained from the quadratic factor, corresponds to the correct answer B) 2.