Proof Problem #2

If $a, b$ are the roots of $x^4 + x^3 - 1 = 0$ then prove that $ab$ is a root of $x^6 + x^4 + x^3 - x^2 - 1 = 0$

#Algebra

Easy Math Editor

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Comments

Suppose $a,b,c,d$ are the four roots of $f(x) = x^{4} + x^{3} - 1.$ Then $f(x) = (x - a)(x - b)(x - c)(x - d) =$

$x^{4} - (a + b + c + d)x^{3} + (ab + ac + ad + bc + bd + cd)x^{2} - (abc + abd + acd + bcd)x + abcd.$

Comparing like coefficients gives us that $a + b + c + d = -1, ab + ac + ad + bc + bd + cd = 0,$

$abc + abd + acd + bcd = 0$ and $abcd = -1.$

Now look at the function $g(x) = (x - ab)(x - ac)(x - ad)(x - bc)(x - bd)(x - cd).$

When we expand this expression, we can use the equations involving $a,b,c,d$ above to establish numerical values for the coefficients of $g(x).$ For example, the coefficient for $x^{5}$ will be $-(ab + ac + ad + bc + bd + cd) = 0,$ and the constant term, i.e., the coefficient of $x^{0}$ , will be $(abcd)^{3} = -1.$ Calculating the other coefficients is quite tedious, (and a bit tricky), but in doing so one will find that $g(x) = x^{6} + x^{4} + x^{3} - x^{2} - 1,$ and so the product of any two of the roots of the first equation will be a root of the second equation.

Had you already got this far and were having difficulty calculating the coefficients, or is this enough of a hint to help you finish the proof?

Brian Charlesworth - 5 years, 8 months ago

Nice suggestion...

Dev Sharma - 5 years, 8 months ago

@Dev Sharma A and b are not the only two roots of the equation.

Mehul Arora - 5 years, 8 months ago

a and b are two roots. But it has other two root also

Dev Sharma - 5 years, 8 months ago

Yes, but you should mention that. You should phrase the question like a and b are two roots of the equation........

Mehul Arora - 5 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$