Real-coefficient polynomial with roots 2,… | Answer

Subject

গণিত

Topic

Algebra

Real-coefficient polynomial with roots 2, 1 ± i; write the monic cubic.
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ক)

x3− 4x2+ 6x − 4

খ)

x3− 4x2+ 5x − 2

গ)

x3− 3x2+ 4x − 2

ঘ)

x3− 2x2+ 2x − 2

Explanation

We are given roots

2,\;1+i,\;1-i

and asked for the monic cubic polynomial with real coefficients. Since complex roots come in conjugate pairs for real-coefficient polynomials, the quadratic factor from

1\pm i

(x-(1+i))(x-(1-i))=((x-1)-i)((x-1)+i)=(x-1)^2+1 = x^2-2x+2.

Multiplying by the linear factor for the root

2

gives the cubic:

(x-2)(x^2-2x+2).

Expand:

\begin{aligned} (x-2)(x^2-2x+2) &= x^3-2x^2+2x -2x^2+4x-4\\ &= x^3-4x^2+6x-4. \end{aligned}

So the monic cubic is

x^3-4x^2+6x-4

, which matches Option 1.

For ax2+ bx + c = 0 with roots r1, r2, state Vieta’s relations.For ax3+ bx2+ cx + d with roots r1, r2, r3, state Vieta’s relations.From x2− 5x + 6 = 0, find the roots using Vieta.Given roots 1 and −3 for a monic quadratic, find the polynomial.

Real-coefficient polynomial with roots 2, 1 ± i; write the monic cubic.Reading

Real-coefficient polynomial with roots 2, 1 ± i; write the monic cubic.

Explanation

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Real-coefficient polynomial with roots 2, 1 ± i; write the monic cubic.
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