11.3 – Rational 11.3 – Rational Functions Functions
11.3 – Rational Functions. Rational functions – Rational functions – algebraic fraction.
Dec 17, 2015
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11.3 – Rational Functions11.3 – Rational Functions
Rational functionsRational functions – –
Rational functionsRational functions – algebraic fraction – algebraic fraction
Rational functionsRational functions – algebraic fraction; – algebraic fraction; numerator and (esp.) denominator are numerator and (esp.) denominator are polynomialspolynomials
Rational functionsRational functions – algebraic fraction; – algebraic fraction; numerator and (esp.) denominator are numerator and (esp.) denominator are polynomialspolynomials
Ex.Ex. Simplify. Simplify.a. a. 3535yzyz22
1414yy22zz
Rational functionsRational functions – algebraic fraction; – algebraic fraction; numerator and (esp.) denominator are numerator and (esp.) denominator are polynomialspolynomials
Ex.Ex. Simplify. Simplify.a. a. 3535yzyz22
1414yy22zz7 7 · 5 · · 5 · y y ·· z z ·· z z
Rational functionsRational functions – algebraic fraction; – algebraic fraction; numerator and (esp.) denominator are numerator and (esp.) denominator are polynomialspolynomials
Ex.Ex. Simplify. Simplify.a. a. 3535yzyz22
1414yy22zz7 7 · 5 · · 5 · y y ·· z z ·· z z7 7 · 2 · · 2 · y y ·· y y ·· z z
Rational functionsRational functions – algebraic fraction; – algebraic fraction; numerator and (esp.) denominator are numerator and (esp.) denominator are polynomialspolynomials
Ex.Ex. Simplify. Simplify.a. a. 3535yzyz22
1414yy22zz7 7 · 5 · · 5 · y y ·· z z ·· z z7 7 · 2 · · 2 · y y ·· y y ·· z z
Rational functionsRational functions – algebraic fraction; – algebraic fraction; numerator and (esp.) denominator are numerator and (esp.) denominator are polynomialspolynomials
Ex.Ex. Simplify. Simplify.a. a. 3535yzyz22
1414yy22zz7 7 · · 55 · · y y ·· zz ·· z z7 7 · · 22 · · y y ·· yy ·· z z
Rational functionsRational functions – algebraic fraction; – algebraic fraction; numerator and (esp.) denominator are numerator and (esp.) denominator are polynomialspolynomials
Ex.Ex. Simplify. Simplify.a. a. 3535yzyz22
1414yy22zz7 7 · · 55 · · y y ·· zz ·· z z7 7 · · 22 · · y y ·· yy ·· z z55zz22yy
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2)( + 2)(xx + 4) + 4)
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2)( + 2)(xx + 4) + 4)
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4) 11 xx + 2 + 2
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
11 ; ; x ≠ -2x ≠ -2
xx + 2 + 2
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
11 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4
xx + 2 + 2
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
11 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4
xx + 2 + 2
c. c. xx2 2 – – xx – 20 – 20xx22 + 6 + 6xx + 8 + 8
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
11 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4
xx + 2 + 2
c. c. xx2 2 – – xx – 20 – 20xx22 + 6 + 6xx + 8 + 8
((xx – 5)( – 5)(xx + 4) + 4)
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
11 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4
xx + 2 + 2
c. c. xx2 2 – – xx – 20 – 20xx22 + 6 + 6xx + 8 + 8
((xx – 5)( – 5)(xx + 4) + 4) ((xx + 2)( + 2)(xx + 4) + 4)
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
11 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4
xx + 2 + 2
c. c. xx2 2 – – xx – 20 – 20xx22 + 6 + 6xx + 8 + 8
((xx – 5) – 5)((xx + 4) + 4) ((xx + 2) + 2)((xx + 4) + 4)
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
11 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4
xx + 2 + 2
c. c. xx2 2 – – xx – 20 – 20xx22 + 6 + 6xx + 8 + 8
((xx – 5)( – 5)(xx + 4) + 4) ((xx + 2)( + 2)(xx + 4) + 4)
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
11 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4
xx + 2 + 2
c. c. xx2 2 – – xx – 20 – 20xx22 + 6 + 6xx + 8 + 8
((xx – 5) – 5)((xx + 4) + 4) ((xx + 2) + 2)((xx + 4) + 4)
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4)
11 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4
xx + 2 + 2
c. c. xx2 2 – – xx – 20 – 20xx22 + 6 + 6xx + 8 + 8
((xx – 5) – 5)((xx + 4) + 4) ((xx + 2) + 2)((xx + 4) + 4)
xx – 5 – 5 ; ; x x ≠ -2≠ -2
xx + 2 + 2
b. b. xx + 4 + 4xx22 + 6 + 6xx + 8 + 8 xx + 4 + 4((xx + 2) + 2)((xx + 4) + 4) 11 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4 xx + 2 + 2
c. c. xx2 2 – – xx – 20 – 20xx22 + 6 + 6xx + 8 + 8
((xx – 5) – 5)((xx + 4) + 4) ((xx + 2) + 2)((xx + 4) + 4)
xx – 5 – 5 ; ; x x ≠ -2, ≠ -2, x x ≠ -4≠ -4xx + 2 + 2
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