Exercise 2.2

1. $$(x-4)(x+2);\quad 4,-2;\quad \alpha+\beta=2=-\frac ba,\ \alpha\beta=-8=\frac ca$$
2. $$(2s-1)^2;\quad \frac12,\frac12;\quad \alpha+\beta=1=-\frac ba,\ \alpha\beta=\frac14=\frac ca$$
3. $$(3x+1)(2x-3);\quad -\frac13,\frac32;\quad \alpha+\beta=\frac76=-\frac ba,\ \alpha\beta=-\frac12=\frac ca$$
4. $$4u(u+2);\quad 0,-2;\quad \alpha+\beta=-2=-\frac ba,\ \alpha\beta=0=\frac ca$$
5. $$(t-\sqrt{15})(t+\sqrt{15});\quad \sqrt{15},-\sqrt{15};\quad \alpha+\beta=0=-\frac ba,\ \alpha\beta=-15=\frac ca$$
6. $$(3x-4)(x+1);\quad \frac43,-1;\quad \alpha+\beta=\frac13=-\frac ba,\ \alpha\beta=-\frac43=\frac ca$$

For sum S and product P, one suitable polynomial is x² − Sx + P. Multiplying by a non-zero constant gives an equivalent answer.

1. $$4x^2-x-4$$
2. $$3x^2-3\sqrt2x+1$$
3. $$x^2+\sqrt5$$
4. $$x^2-x+1$$
5. $$4x^2+x+1$$
6. $$x^2-4x+1$$