6. Let ao, a1, a2, a3, a4 be constant real numbers such that a1 a2 a3 + + + + a4 = 0. 5 4 3 2 Show that the polynomial P(x) = ax² + a₁x³ + a₂x² +a3x+a4 has at least one zero between 0 and 1. (Hint: Consider a1 4 a2 5 a3 f(x) = 2x³ + 2/1x²¹ +23²x³² +22³x² + Show that Rolle's theorem applies to f(x) on the interval [0, 1]. Deduce that P(x) has a 0 in [0, 1]).

6. List all polynomials of the form x++bx+cx²+dx+e with rational coefficients that are factors of the polynomial, x¹²-1.

9. Find all real zeroes of the function p(x)= 6x' +17x – 26x +8 Solution: https://youtu.be/XBNPBY4SAB8 10. Construct a polynomial with real coefficients that has zeroes 2 and 3+i. Solution: https://youtu.be/J9c9czrqFJA 11. Find all real/complex zeroes and write the polynomial in factored form. p(x) = 3x* – 29x +101x² – 119x – 52 Solution: https://youtu.be/78Tdn3Y0dj4 x- 4 12. Express the domain in interval notation: R(x)=- x+7 %3D

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