91Ó°ÊÓ

(a) Compute the discriminant of the quadratic and note that it is negative (and therefore the equation has no real-number roots). (b) Use the quadratic formula to obtain the two complex conjugate roots of each equation. $$3 z^{2}-7 z+5=0$$

If \(p\) and \(q\) are prime numbers, show that the equation \(x^{3}+p x-p q=0\) has no rational roots.

Given that the identity \(f(t)=d(t) \cdot q(t)+R(t)\) holds for the following polynomials, evaluate \(f(4)\) $$f(t)=t^{5}-3 t^{4}+2 t^{3}-5 t^{2}+6 t-7$$ $$d(t)=t-4$$ $$q(t)=t^{4}+t^{3}+6 t^{2}+19 t+82$$ $$R(t)=321$$

Find the remainder when \(t^{5}-5 a^{4} t+4 a^{5}\) is divided by \(t-a\).

Find all integral values of \(b\) for which the equation \(x^{3}-b^{2} x^{2}+3 b x-4=0\) has a rational root.

(a) Compute the discriminant of the quadratic and note that it is negative (and therefore the equation has no real-number roots). (b) Use the quadratic formula to obtain the two complex conjugate roots of each equation. $$\frac{1}{6} z^{2}-\frac{1}{4} z+1=0$$

One root of the equation \(x^{2}+b x+1=0\) is twice the other; find \(b .\) (There are two answers.)

When \(f(x)\) is divided by \((x-a)(x-b),\) the remainder is \(A x+B .\) Apply the division algorithm to show that $$A=\frac{f(a)-f(b)}{a-b} \quad \text { and } \quad B=\frac{b f(a)-a f(b)}{b-a}$$

Determine a value for \(a\) such that one root of the equation \(a x^{2}+x-1=0\) is five times the other.

(a) Compute the discriminant of the quadratic and note that it is negative (and therefore the equation has no real-number roots). (b) Use the quadratic formula to obtain the two complex conjugate roots of each equation. $$\frac{1}{2} z^{2}+2 z+\frac{9}{4}=0$$