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