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Difficulty: MediumQuadratic Equations and Polynomial Factoring

If x=1x = 1 is a root of the cubic polynomial P(x)=x3+ax2−11x+12P(x) = x^3 + ax^2 - 11x + 12, where aa is a constant, what is the value of the largest real root of P(x)P(x)?

Answer: 4

Answer

The largest real root of the polynomial P(x)P(x) is 44.
Evaluating P(1)=0P(1) = 0 yields 1+a−11+12=01 + a - 11 + 12 = 0, giving a=−2a = -2. Factoring the polynomial x3−2x2−11x+12x^3 - 2x^2 - 11x + 12 by (x−1)(x - 1) produces the quadratic expression x2−x−12x^2 - x - 12. Factoring this quadratic gives (x−4)(x+3)=0(x - 4)(x + 3) = 0. The roots of the polynomial are −3-3, 11, and 44. Comparing these values, the maximum root is 44.

Step-by-Step Solution

1
Find the constant aa using the given root x=1x = 1.
a=−2a = -2
Since x=1x = 1 is a root, P(1)=13+a(1)2−11(1)+12=0P(1) = 1^3 + a(1)^2 - 11(1) + 12 = 0, which simplifies to a+2=0a + 2 = 0.
2
Factor (x−1)(x - 1) out of P(x)=x3−2x2−11x+12P(x) = x^3 - 2x^2 - 11x + 12.
P(x)=(x−1)(x2−x−12)P(x) = (x - 1)(x^2 - x - 12)
Dividing x3−2x2−11x+12x^3 - 2x^2 - 11x + 12 by (x−1)(x - 1) yields the quotient x2−x−12x^2 - x - 12.
3
Factor the quadratic factor x2−x−12x^2 - x - 12.
(x−4)(x+3)(x - 4)(x + 3)
The quadratic factors into two linear terms whose roots are 44 and −3-3.
4
Determine the largest of all real roots of P(x)P(x).
44
The complete set of real roots is {−3,1,4}\{-3, 1, 4\}, of which 44 is the greatest.

Key Concept

Polynomial Factoring and Remainder/Factor Theorem
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Topics

Quadratic Equations and Polynomial Factoring

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