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

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.

Return to Practice (Potential Rational Roots).