Find the rational roots using the rational zero theorem.

x4−10x2+9

Question

Find the rational roots using the rational zero theorem.

x4−10x2+9

BlackTom AI · Accepted Answer

The given polynomial is x^4 − 10x^2 + 9, and we are asked to find its rational roots using the Rational Root Theorem.
First, notice that the polynomial is in even powers of x, so it makes sense to substitute t = x^2. This turns the equation into a quadratic in t: t^2 − 10t + 9 = 0.
Next, solve the quadratic for t: the discriminant is Δ = (−10)^2 − 4(1)(9) = 100 − 36 = 64, so t = [10 ± √64]/2 = [10 ± 8]/2.
This yields two possible values for t: t = 1 and t = 9.
Since t = x^2, translate back to x: x^2 = 1 or x^2 = 9.
From x^2 = 1 we get x = ±1, and from x^2 = 9 we get x = ±3.
Therefore, the rational roots are the numbers whose squares are 1 or 9, corresponding to x = −3, −1, 1, 3.
In the context of the Rational Root Theorem, the possible rational roots to test would be factors of the constant term (±1, ±3, ±9) over factors of the leading coefficient (±1). The factorization above shows which of these actually satisfy the equation, via the t-substitution route.

ALGII141-027 4.7 Practice Problems

Find the rational roots using the rational zero theorem. x4−10x2+9

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