# POLQ 3 | Lesson 2 | Practice (Finding Rational Roots) Solutions

Solutions:

1) $$f(x)=x^2+8x+10$$

The potential rational roots are $$\pm 1$$, $$\pm 2$$, $$\pm 5$$

2) $$f(x)=x^2-64$$

The potential rational roots are $$\pm 1$$, $$\pm 2$$, $$\pm 4$$, $$\pm 8$$, $$\pm 16$$, $$\pm 32$$, $$\pm 64$$. Factoring reveals the actual rational roots are $$\pm 8$$.

3) $$f(x)=5x^3-2x^2+20x-8$$

The potential rational roots are $$\pm 1$$, $$\pm 2$$, $$\pm 4$$, $$\pm 8$$, $$\pm\Large\frac{1}{5}$$, $$\pm \Large\frac{2}{5}$$, $$\pm \Large\frac{4}{5}$$, $$\pm\Large\frac{8}{5}$$.

For the problems below, find the possible rational roots and then use synthetic division and other factoring strategies to find all the rational roots of the functions.

1) $$f(x)=x^3+x^2-5x+3$$

The potential rational roots are  $$\pm 1$$, $$\pm 3$$.

1 is a root. This can be found through synthetic division, or by evaluating the function for 1.

$$f(1)= 1^3+ 1^2 -5•1 + 3$$; or $$f(1) = 0$$

Dividing out $$(x-1)$$, the resulting quadratic is $$x^2+2x-3$$. Standard factoring for a quadratic will help you find the other potential rational roots which are 1 and 3. The root of 1 is said to have a multiplicity of 2.

2) $$f(x)=x^3-13x^2+23x-11$$

The potential rational roots are $$\pm 1$$, $$\pm 11$$. One is also a root of this polynomial. Remaining is the quadratic $$x^2-12x+11$$. When factored the other two rational roots are 1 and 11. This polynomial also has a root of one with multiplicity of 2.