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How Do You Find Excluded Values?

Find the excluded values of x for the given rational expression.

Summary

  1. Excluded values are the values of a variable that make a rational expression undefined
  2. The excluded values are the values for 'x' that will make the denominator, x2 - 25, equal 0
  3. To find the excluded values, set the denominator equal to 0 and solve for 'x'
  4. In order for x2 - 25 = 0 to be true, x2 must be equal to 25
  5. Subtracting two things that are squared means that we have a difference of squares
  6. Use the Zero Product Property to split the equation up and solve each equation for 'x'
  7. 5 and -5 are the excluded values for our rational expression

Notes

    1. Excluded values are the values of a variable that make a rational expression undefined
    1. A rational expression is a polynomial divided by another polynomial
    1. A rational expression is just like a big fraction, with polynomials instead of numbers in the numerator and denominator
    2. Remember, we can't divide by 0!
    3. So if we have a value for a variable that makes the denominator equal 0, that fraction will be undefined
    1. If the denominator equals 0, then the rational expression will be undefined
    2. So any values of 'x' that make the denominator, x2 - 25, equal 0 need to be excluded
    3. When we exclude values, we're basically just saying that 'x' cannot equal those values
    4. So we're putting some restrictions on what 'x' can be
    1. In order to figure out which values need to be excluded, we need to know which values for 'x' make the denominator 0
    2. So to do that, we can set the denominator equal to 0 and solve for 'x'
    3. The answers we get when we solve that equation will be the values that 'x' CANNOT be
    1. The values for 'x' that we get when we solve x2 - 25 = 0 will be the excluded values
    1. In order for x2 - 25 = 0 to be true, x2 must be equal to 25, since 25 - 25 = 0
    2. So we need to have 25 be something squared
    3. This is the same as finding the square root of 25
    4. But remember, we get a positive number when we square both positive AND negative numbers
    5. So 25 could either be 52 OR (-5)2
    1. If 'x' was either 5 or -5, we would get 25 - 25 = 0 when we plugged in
    2. This would make our denominator 0, so we would need to exclude those values for 'x'
    3. We could solve this equation by adding 25 on both sides and taking the square root
    4. But we could also factor this equation and solve using the Zero Product Property
    1. The left hand side of our equation looks like it will factor nicely, so let's try that
    1. 25 is a perfect square, which means we can rewrite it as an integer squared
    2. The square root of 25 is 5, which means that 52 equals 25
    1. Since we know our expression is a difference of squares, we can use a shortcut to factor it
    1. When we multiply two binomials together and get 0, we know that at least one of them must equal 0
    2. So to get the possible values for 'x', we can set each binomial equal to 0 and solve each equation separately
    1. 5 and -5 are the values for 'x' that will make x2 - 25 equal 0
    2. Before when we tried to figure out what values for 'x' would give us 25 - 25 = 0, we also came up with 5 and -5
    1. If we plug 5 or -5 in for 'x' in our rational expression, the denominator will become 0
    2. Since we can't have 0 in the denominator of a fraction, we can't have x equal 5 or -5
    3. So we call 5 and -5 our 'excluded values'