Polynomials and rational expressions
Any spelling of the right answer counts: 0.5 or 1/2, pi/2 or π/2, sine or cosine for the same graph.
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About Polynomials and rational expressions
Dividing a polynomial by x − r works like long division with numbers, and synthetic division is the quick version. The last number it leaves is the remainder, and the remainder theorem says that number is just p(r). When the remainder is 0, x − r is a factor.
A rational expression is a fraction of polynomials. Simplify it the way you simplify 6/8: factor the top and the bottom and cancel what they share. The expression is undefined wherever the bottom is 0, even if that factor cancels.
Keep in mind
- Remainder theorem: dividing p(x) by x − r leaves the remainder p(r).
- Factor theorem: x − r is a factor exactly when p(r) = 0.
- Synthetic division: bring down, multiply by r, add, repeat. The last number is the remainder.
- To simplify a rational expression, factor the top and bottom and cancel shared factors.
- The expression is undefined where the bottom equals 0.
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