2. Radical Equations and Problem Solving 4.7

3. Power Rule When solving radical equations, we use a new principle called the power rule. –The Power Rule states that if a = b, then a n = b n and vice versa. This also applies to radicals, specifically: –If a = b, then.

4. Example 1 If a radical with a variable is already isolated in an equation, we will raise each side of the equation to a power that matches the radical index. Solve the radical equations below.

5. Extraneous Solutions As was the case with rational equations, it was necessary to check answers to be sure they verified the original equation. With radical equations, it is also necessary for two primary reasons: –With an even index, we cannot have a negative radicand (i.e. we cannot take the square root of a negative) –Unless specified otherwise, we will only calculate the principal root of any radical. Example 2: Solve and check your solutions.

6. Solving polynomial inequalities Rewrite the polynomial so that all terms are on one side and zero on the other. Factor the polynomial. We are interested in when factors are either pos. or neg., so we must know when the factor equals zero. The values of x for which the factors equal zero are the boundary points, which we place on the number line. The intervals around the boundary points must be tested to find on which interval(s) will the polynomial be positive/negative.

7. Solve : (x – 3)(x + 1)(x – 6) < 0 To solve this inequality we observe that 0 is already on one side and the polynomial is factored already on the other side. The 3 boundary values are x = 3,-1,6 They create 4 intervals: Pick a number in each interval to test the sign of that interval. If the polynomial is negative there then the interval is in the solution set. Solution set:

8. Solve: x 3 +3x 2 ≥ 10x 1.To solve, first we must rewrite the inequality so all terms are on one side and 0 on the other, then factor. 2.x(x-2)(x+5) ≥ 0 3.Boundary points: 0, 2, -5 4.Solution set:

9. Solving rational inequalities VERY similar to solving polynomial inequalites EXCEPT if the denominator equals zero, there is a domain restriction. The function COULD change signs on either side of that point. Step 1: Rewrite the inequality so all terms are on one side and zero on the other. Step 2: Factor both numerator & denominator to find boundary values for regions to check when function becomes positive or negative. And do as before !

10. Solve the following inequalities: 1) 2) 3)