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2. A rational function is a function of the form: (x) - p(x) - q(x)' where p(x) and g(x) are ordinary polynomials with real coefficients, with no common factor of degree 1 or more. Note that this includes the possibility that g(x) = c, a constant, in which case r(x) is just a polynomial. This is analogous to the integers Z being a subset of the rationals Q. We define addition of two rational functions and scalar multiplication by a real number as we do in Algebra: k.p(x) Pi(x) 91(x) p2(x) pi(x) • 92(x) + p₂(x) • 9₁(x) 92(x) 91(x)q2(x) " and ko p(x) g(x) The final answer may be reduced by cancelling out common factors in the numerator and denominator, like we do when we perform operations on ordinary fractions. compute-32x+24 and-35x-7 x + 3 As a warm-up, x²-9 6x +9° Let R be the set of all rational functions. Show that R is a vector space under this addition and scalar multiplication.

2. A rational function is a function of the form: (x) - p(x) - q(x)' where p(x) and g(x) are ordinary polynomials with real coefficients, with no common factor of degree 1 or more. Note that this includes the possibility that g(x) = c, a constant, in which case r(x) is just a polynomial. This is analogous to the integers Z being a subset of the rationals Q. We define addition of two rational functions and scalar multiplication by a real number as we do in Algebra: k.p(x) Pi(x) 91(x) p2(x) pi(x) • 92(x) + p₂(x) • 9₁(x) 92(x) 91(x)q2(x) " and ko p(x) g(x) The final answer may be reduced by cancelling out common factors in the numerator and denominator, like we do when we perform operations on ordinary fractions. compute-32x+24 and-35x-7 x + 3 As a warm-up, x²-9 6x +9° Let R be the set of all rational functions. Show that R is a vector space under this addition and scalar multiplication.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
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2. A rational function is a function of the form: r(x) - p(x) q(x)' where p(x) and g(x) are ordinary polynomials with real coefficients, with no common factor of degree 1 or more. Note that this includes the possibility that g(x) = c, a constant, in which case r(x) is just a polynomial. This is analogous to the integers Z being a subset of the rationals Q. Pi(x) 91(x) We define addition of two rational functions and scalar multiplication by a real number as we do in Algebra: p2(x) pi(x) • 92(x) + p2(x) • 9₁(x) 92(x) 91(x)q2(x) " and ko p(x) g(x) k.p(x) g(x) The final answer may be reduced by cancelling out common factors in the numerator and denominator, like we do when we perform operations on ordinary fractions. compute-32x+24 and-35x-7 x + 3 As a warm-up, x²-9 6x +9° Let R be the set of all rational functions. Show that R is a vector space under this addition and scalar multiplication.
Transcribed Image Text:2. A rational function is a function of the form: r(x) - p(x) q(x)' where p(x) and g(x) are ordinary polynomials with real coefficients, with no common factor of degree 1 or more. Note that this includes the possibility that g(x) = c, a constant, in which case r(x) is just a polynomial. This is analogous to the integers Z being a subset of the rationals Q. Pi(x) 91(x) We define addition of two rational functions and scalar multiplication by a real number as we do in Algebra: p2(x) pi(x) • 92(x) + p2(x) • 9₁(x) 92(x) 91(x)q2(x) " and ko p(x) g(x) k.p(x) g(x) The final answer may be reduced by cancelling out common factors in the numerator and denominator, like we do when we perform operations on ordinary fractions. compute-32x+24 and-35x-7 x + 3 As a warm-up, x²-9 6x +9° Let R be the set of all rational functions. Show that R is a vector space under this addition and scalar multiplication.
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