The following polynomial has integer roots $x^{3} + 30 x^{2} - 7377 x + 14626 = 0$ . Find the value of the largest root.

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — 71

Detailed Solution

For this monic cubic the roots have product $- 14626$ and sum $- 30$ . Factorising, $14626 = 2 \times 71 \times 103$ .

Try the roots $2$ , $71$ , $- 103$ : the sum is $2 + 71 - 103 = - 30$ and the product is $2 \cdot 71 \cdot (- 103) = - 14626$ .

The pair sum is $2 \cdot 71 + 2 (- 103) + 71 (- 103) = 142 - 206 - 7313 = - 7377$ , as required. The largest root is 71.

✓ Check

71^{3} + 30 \cdot 71^{2} - 7377 \cdot 71 + 14626 = 357911 + 151230 - 523767 + 14626 = 0

⚠ Common trap

The root 103 has the largest size, but its sign is negative (the roots add to

- 30

), so the largest root is 71; 103 is not even an option.

Quick tip

$14626 = 2 \times 71 \times 103$ ; roots $2, 71, - 103$

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Algebraic Polynomials & Identities — Preparation

This question tests your understanding of Algebraic Polynomials & Identities, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A polynomial is an algebraic expression made of terms with non-negative whole-number powers of a variable. This topic covers the degree and zeros of a polynomial, the remainder and factor theorems, factorisation, and standard algebraic identities such as (a + b)^2, (a - b)^2, a^2 - b^2, (a + b)^3, a^3 + b^3 and a^3 + b^3 + c^3 - 3abc. Identities are among the biggest time-savers in aptitude. Questions that seem to need heavy calculation, such as evaluating x^3 + 1/x^3 when x + 1/x is known, become one-line answers when the right identity is used. The remainder theorem similarly lets you find a remainder without long division: the remainder when p(x) is divided by (x - a) is p(a). Practise the polynomial and identity questions below, each with a step-by-step solution.

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The following polynomial has integer roots x3+30x2−7377x+14626=0. Find the value of the largest root.

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Algebraic Polynomials & Identities — Preparation

The following polynomial has integer roots $x^{3} + 30 x^{2} - 7377 x + 14626 = 0$ . Find the value of the largest root.