Question Video: Roots of Cubics | Nagwa Question Video: Roots of Cubics | Nagwa

# Question Video: Roots of Cubics Mathematics

Given that π is one of the roots of the equation π₯Β³ β 5π₯Β² + π₯ β 5 = 0, find the other two roots.

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### Video Transcript

Given that π is one of the roots of the equation π₯ cubed minus five π₯ squared plus π₯ minus five equals zero, find the other two roots.

The factor theorem tells us that π₯ minus π is one of the factors of this polynomial. And so, we could divide the left-hand side by this factor to get the quadratic ππ₯ squared plus ππ₯ plus π, which we could then solve. But we can make things easier for ourselves by using the conjugate root theorem which tells us that the complex conjugate of π must also be a root as weβre dealing with a polynomial with real coefficients. And so, π₯ plus π again by the factor theorem must be a factor of this polynomial.

Multiplying the two known factors together, we get π₯ squared plus one. And we can distribute again. And comparing coefficients, we can see that π is one and π is negative five. We can substitute these values then, factoring our cubic as π₯ minus one times π₯ plus one times π₯ minus five. Remember that weβre looking for the roots of this equation. And we can just read them off from the factored form. We find that they are five, negative π, and π. For this problem, the conjugate root theorem saved us some working, but wasnβt essential.

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