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Factorize the polynomials in the denominator and numerator. |
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| 2. |
Cancel the common factors between the numerator and denominator. |
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There are some important formulas for factorizing polynomials. Just like for quadratic equation we know that (x+a)2 = x2+a2+2ax. Some of these formulas are given below: |
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We can check these formulas by multiplying the factors using the distributive property. |
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i.
ii.
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(x+a)3 = x3 + a3 + 3x2a + 3ax2
(x-a)3 = x3 - a3 - 3x2a + 3ax2
(x3+a3) = (x+a)(x2 + a2 - ax) |
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Let’s try to prove it: |
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Right hand side = (x+a)(x2 + a2 - ax)
= x3 + xa2 – ax2 + ax2 + a3 – a2x
= x3 + a3 = Left hand side = proof. |
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iv |
(x3 - a3) = (x - a)(x2 + a2 + ax) |
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Example 1: |
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Simplify . |
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Solution |
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Solving Rational Equations and Inequalities Algebraically |
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We follow the same steps as mentioned above to simplify the rational expression. |
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The simplified expression is then solved to find the value of x in the equation given by r1(x) = r2(x) or r(x) = a where r(x), r1(x) and r2(x) are any rational expressions and ‘a’ is a constant. We use the cross multiplication method to solve the equation. |
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Example : 1 |
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1. The expressions are already simplified. |
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2. Cross multiply, i.e. multiply the numerator of the left side by the denominator of the right side = the numerator of the right side multiplied by the denominator of the left side. |
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- 3x=4x+4
- Move the variables to the left side and the constants to the right side by using inverse operations.
- Subtract 4x from both the sides.
- 3x-4x=4
- -x = 4 or x = -4.
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Example : 2 |
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The same question above can be changed to an inequality as shown: |
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We use the same steps as for example 1: |
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3x ≥ 4x+4 |
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- -x ≥ 4
- Multiplying both the sides with -1, x ≤ -4.
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Try these problems |
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| 1. |
Solve the rational equation 9/28 + 3/(z + 2) = 3 / 4. |
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A.
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-5
4
-4
5 |
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Answer: D. |
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| 2. |
Simplify the following equation: |
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Answer: |
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The numerator can be factored as (x+1)(x-1). |
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| Thus, the answer = |
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