Question 1127820
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The rational roots theorem says that the possible rational roots for a polynomial are +/-(p/q), where p is a factor of the constant term and q is a factor of the leading coefficient.  For this polynomial then, the possible roots are<br>
+/- {1, 2, 4, 1/2}<br>
When you find one root, you can divide the polynomial by the corresponding linear factor to obtain a quadratic polynomial; you can then find the other two roots by factoring the quadratic or using the quadratic formula.<br>
So you only need to find one root using the rational roots theorem.  Which ones should you try first?<br>
1 and -1 are always the easiest to test, simply by evaluating the polynomial for those values.  In this example, neither 1 nor -1 is a root.<br>
For the other possible rational roots, some students will prefer evaluating the polynomial; other students will find synthetic division easier.  Try the smaller integers (positive and negative) first.  And hope that you find a root before you need to test the fractions.<br>
In this example, the polynomial evaluated at x=2 is 0, so 2 is a root.  Dividing the polynomial by the linear factor (x-2) using synthetic division shows the remaining polynomial to be 2x^2+5x+2.  That is easily factored to finish the factorization of the polynomial.