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Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = x^2 - 5x + 3. Compute a^3*b + a*b^3.

Dec 13, 2022

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$$ab^3 + a^3b = ab(a^2 + b^2) = ab((a+b)^2 - 2ab)$$

$$x^2 - 3x +4 = 0$$

$$a + b = -{b \over a} = 3$$

$$ab = {c\over a} = 4$$

$$ab^3 + a^3b = ab((a+b)^2 - 2ab) = 4(9 - 8) = \color{brown}\boxed 4$$

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Dec 13, 2022