asymptotes of (x^2+4x+3)/(x^2-1)

{ "query": { "display": "asymptotes $$\\frac{x^{2}+4x+3}{x^{2}-1}$$", "symbolab_question": "FUNCTION#asymptotes \\frac{x^{2}+4x+3}{x^{2}-1}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Vertical}: x=1,\\mathrm{Horizontal}: y=1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{x^{2}+4x+3}{x^{2}-1}:\\quad\\:$$Vertical$$:\\:x=1,\\:$$Horizontal$$:\\:y=1$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{x^{2}+4x+3}{x^{2}-1}:{\\quad}x=1$$", "input": "\\frac{x^{2}+4x+3}{x^{2}-1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{x^{2}+4x+3}{x^{2}-1}:{\\quad}\\frac{x+3}{x-1}$$", "input": "\\frac{x^{2}+4x+3}{x^{2}-1}", "steps": [ { "type": "interim", "title": "Factor $$x^{2}+4x+3:{\\quad}\\left(x+1\\right)\\left(x+3\\right)$$", "input": "x^{2}+4x+3", "result": "=\\frac{\\left(x+1\\right)\\left(x+3\\right)}{x^{2}-1}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}+4x+3", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=4,\\:c=3$$", "$$u*v=3,\\:u+v=4$$" ] }, { "type": "interim", "title": "Factors of $$3:{\\quad}1,\\:3$$", "input": "3", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$3:{\\quad}3$$", "input": "3", "steps": [ { "type": "step", "primary": "$$3$$ is a prime number, therefore no factorization is possible", "result": "=3" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRstjW1xhuS30EMAtGsK3e/Pwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJvvtJxbo6u/80Nt/bk1au5a/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$3$$", "result": "1,\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=3,\\:$$check if $$u+v=4$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=3:\\quad\\:u*v=3,\\:u+v=4\\quad\\Rightarrow\\quad\\:$$True" } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=1,\\:v=3" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}+x\\right)+\\left(3x+3\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwuEndPnjUh16y5iapC5Yr/MsjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKG3UwAYNOt7FBX9IfTXfsvAJUs6AZ1NJ9ukxL4iA4UZnEQbxpAQHC97Q3bEFYSDINpkfHD58K/uguquJtFZGMAOkb6n80yx79/CGv9RvUIw1A==" } }, { "type": "step", "result": "=\\left(x^{2}+x\\right)+\\left(3x+3\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}+x:\\quad\\:x\\left(x+1\\right)$$", "input": "x^{2}+x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx+x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$3\\:$$from $$3x+3:\\quad\\:3\\left(x+1\\right)$$", "input": "3x+3", "steps": [ { "type": "step", "primary": "Factor out common term $$3$$", "result": "=3\\left(x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HbG+7Qsnh5yQsO5z+Ku/c8evWYjE5TzvOKvzjFpIz0IByGqvpVrcovBtOEfrP9G1YkMBvNfXtkhCqU5k1KRx86N6Hv6MoTMtvtU0IQwXdn/LaG2/wGnl6dNw3H9m17XsqgP90/iWrLaqXlriAxgSFT9N4sQNNgVLtD4hZOImHZY=" } }, { "type": "step", "result": "=x\\left(x+1\\right)+3\\left(x+1\\right)" }, { "type": "step", "primary": "Factor out common term $$x+1$$", "result": "=\\left(x+1\\right)\\left(x+3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$x^{2}-1:{\\quad}\\left(x+1\\right)\\left(x-1\\right)$$", "input": "x^{2}-1", "result": "=\\frac{\\left(x+1\\right)\\left(x+3\\right)}{\\left(x+1\\right)\\left(x-1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=x^{2}-1^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$x^{2}-1^{2}=\\left(x+1\\right)\\left(x-1\\right)$$" ], "result": "=\\left(x+1\\right)\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$x+1$$", "result": "=\\frac{x+3}{x-1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "definition", "title": "Vertical asymptotes of rational Functions", "text": "For rational functions, the vertical asymptotes are the undefined points, also known as the zeros of the denominator, of the simplified function." }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=1$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{x+3}{x-1}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x-1=0:{\\quad}x=1$$", "input": "x-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "x-1=0", "result": "x=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "x-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "x=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=1" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "The vertical asymptotes are:", "result": "x=1" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeKIxIKVQdyHueL3nzCoYNBb5mQUE77HOsYsBZNbKbo9GwJ5wwhCrpBMlDJOmV/ZqpsfvkzNq6HpGXH3yEXiCOJ5Uvn/K0adIu922CUSTnadgnw9n+TINobSn51ul7HmvHmwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{x^{2}+4x+3}{x^{2}-1}:{\\quad}y=1$$", "input": "\\frac{x^{2}+4x+3}{x^{2}-1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{x^{2}+4x+3}{x^{2}-1}:{\\quad}\\frac{x+3}{x-1}$$", "input": "\\frac{x^{2}+4x+3}{x^{2}-1}", "steps": [ { "type": "interim", "title": "Factor $$x^{2}+4x+3:{\\quad}\\left(x+1\\right)\\left(x+3\\right)$$", "input": "x^{2}+4x+3", "result": "=\\frac{\\left(x+1\\right)\\left(x+3\\right)}{x^{2}-1}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}+4x+3", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=4,\\:c=3$$", "$$u*v=3,\\:u+v=4$$" ] }, { "type": "interim", "title": "Factors of $$3:{\\quad}1,\\:3$$", "input": "3", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$3:{\\quad}3$$", "input": "3", "steps": [ { "type": "step", "primary": "$$3$$ is a prime number, therefore no factorization is possible", "result": "=3" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRstjW1xhuS30EMAtGsK3e/Pwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJvvtJxbo6u/80Nt/bk1au5a/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$3$$", "result": "1,\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=3,\\:$$check if $$u+v=4$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=3:\\quad\\:u*v=3,\\:u+v=4\\quad\\Rightarrow\\quad\\:$$True" } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=1,\\:v=3" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}+x\\right)+\\left(3x+3\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwuEndPnjUh16y5iapC5Yr/MsjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKG3UwAYNOt7FBX9IfTXfsvAJUs6AZ1NJ9ukxL4iA4UZnEQbxpAQHC97Q3bEFYSDINpkfHD58K/uguquJtFZGMAOkb6n80yx79/CGv9RvUIw1A==" } }, { "type": "step", "result": "=\\left(x^{2}+x\\right)+\\left(3x+3\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}+x:\\quad\\:x\\left(x+1\\right)$$", "input": "x^{2}+x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx+x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$3\\:$$from $$3x+3:\\quad\\:3\\left(x+1\\right)$$", "input": "3x+3", "steps": [ { "type": "step", "primary": "Factor out common term $$3$$", "result": "=3\\left(x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HbG+7Qsnh5yQsO5z+Ku/c8evWYjE5TzvOKvzjFpIz0IByGqvpVrcovBtOEfrP9G1YkMBvNfXtkhCqU5k1KRx86N6Hv6MoTMtvtU0IQwXdn/LaG2/wGnl6dNw3H9m17XsqgP90/iWrLaqXlriAxgSFT9N4sQNNgVLtD4hZOImHZY=" } }, { "type": "step", "result": "=x\\left(x+1\\right)+3\\left(x+1\\right)" }, { "type": "step", "primary": "Factor out common term $$x+1$$", "result": "=\\left(x+1\\right)\\left(x+3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$x^{2}-1:{\\quad}\\left(x+1\\right)\\left(x-1\\right)$$", "input": "x^{2}-1", "result": "=\\frac{\\left(x+1\\right)\\left(x+3\\right)}{\\left(x+1\\right)\\left(x-1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=x^{2}-1^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$x^{2}-1^{2}=\\left(x+1\\right)\\left(x-1\\right)$$" ], "result": "=\\left(x+1\\right)\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$x+1$$", "result": "=\\frac{x+3}{x-1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "definition", "title": "Horizontal asymptotes of rational functions", "text": "If denominator's degree > numerator's degree, the x-axis is the horizontal asymptote.<br/>If the degrees are equal, there is an horizontal asymptote: $$y=\\frac{\\mathrm{numerator's\\:leading\\:coefficient}}{\\mathrm{denominator's\\:leading\\:coefficient}}$$<br/>Otherwise, there is no horizontal asymptote." }, { "type": "step", "primary": "The degree of the numerator$$=1.\\:$$The degree of the denominator$$=1$$", "secondary": [ "The degrees are equal, the horizontal asymptote is: $$y=\\frac{\\mathrm{numerator's\\:leading\\:coefficient}}{\\mathrm{denominator's\\:leading\\:coefficient}}$$" ] }, { "type": "step", "primary": "Numerator's leading coefficient$$=1,\\:$$Denominator's leading coefficient$$=1$$", "result": "y=\\frac{1}{1}" }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=1" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MsbEeQ8t3PpoQbFOMv+1Ci88fRx4krAZo13fswQSt9M8YT3IXPNzJorOYYEGzSCK+VBNmPozZ0OWvMjCTqJf2s1wbhivzD47UeI6IIfI4ahyGfltG457tdWSQCgjCD6OngNUFFSpUtOQxj1BHlqX8G5" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{x^{2}+4x+3}{x^{2}-1}:{\\quad}$$None", "input": "\\frac{x^{2}+4x+3}{x^{2}-1}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{x^{2}+4x+3}{x^{2}-1}:{\\quad}\\frac{x+3}{x-1}$$", "input": "\\frac{x^{2}+4x+3}{x^{2}-1}", "steps": [ { "type": "interim", "title": "Factor $$x^{2}+4x+3:{\\quad}\\left(x+1\\right)\\left(x+3\\right)$$", "input": "x^{2}+4x+3", "result": "=\\frac{\\left(x+1\\right)\\left(x+3\\right)}{x^{2}-1}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}+4x+3", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=4,\\:c=3$$", "$$u*v=3,\\:u+v=4$$" ] }, { "type": "interim", "title": "Factors of $$3:{\\quad}1,\\:3$$", "input": "3", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$3:{\\quad}3$$", "input": "3", "steps": [ { "type": "step", "primary": "$$3$$ is a prime number, therefore no factorization is possible", "result": "=3" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRstjW1xhuS30EMAtGsK3e/Pwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJvvtJxbo6u/80Nt/bk1au5a/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$3$$", "result": "1,\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=3,\\:$$check if $$u+v=4$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=3:\\quad\\:u*v=3,\\:u+v=4\\quad\\Rightarrow\\quad\\:$$True" } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=1,\\:v=3" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}+x\\right)+\\left(3x+3\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwuEndPnjUh16y5iapC5Yr/MsjvX7KVUO/AeCFSId4S337bTjH0JhPfEDoyROh2MmxOR81aIMFSjUznRIvxWbLKG3UwAYNOt7FBX9IfTXfsvAJUs6AZ1NJ9ukxL4iA4UZnEQbxpAQHC97Q3bEFYSDINpkfHD58K/uguquJtFZGMAOkb6n80yx79/CGv9RvUIw1A==" } }, { "type": "step", "result": "=\\left(x^{2}+x\\right)+\\left(3x+3\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}+x:\\quad\\:x\\left(x+1\\right)$$", "input": "x^{2}+x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx+x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$3\\:$$from $$3x+3:\\quad\\:3\\left(x+1\\right)$$", "input": "3x+3", "steps": [ { "type": "step", "primary": "Factor out common term $$3$$", "result": "=3\\left(x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HbG+7Qsnh5yQsO5z+Ku/c8evWYjE5TzvOKvzjFpIz0IByGqvpVrcovBtOEfrP9G1YkMBvNfXtkhCqU5k1KRx86N6Hv6MoTMtvtU0IQwXdn/LaG2/wGnl6dNw3H9m17XsqgP90/iWrLaqXlriAxgSFT9N4sQNNgVLtD4hZOImHZY=" } }, { "type": "step", "result": "=x\\left(x+1\\right)+3\\left(x+1\\right)" }, { "type": "step", "primary": "Factor out common term $$x+1$$", "result": "=\\left(x+1\\right)\\left(x+3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$x^{2}-1:{\\quad}\\left(x+1\\right)\\left(x-1\\right)$$", "input": "x^{2}-1", "result": "=\\frac{\\left(x+1\\right)\\left(x+3\\right)}{\\left(x+1\\right)\\left(x-1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=x^{2}-1^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$x^{2}-1^{2}=\\left(x+1\\right)\\left(x-1\\right)$$" ], "result": "=\\left(x+1\\right)\\left(x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$x+1$$", "result": "=\\frac{x+3}{x-1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "definition", "title": "Slant asymptotes of rational functions", "text": "If numerator's degree = 1 + denominator's degree, there is a slant asymptote of the form: y=mx+b.<br/>Otherwise there is no slant asymptote" }, { "type": "step", "primary": "The degree of the numerator$$=1.\\:$$The degree of the denominator$$=1$$", "secondary": [ "Numerator's degree $$\\neq\\:$$ 1 + denominator's degree" ] }, { "type": "step", "primary": "Therefore there is no slant asymptote" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KL5mikwnGs7EvIxpFPRJ09sNR5nh9tAHlEaYmbcueIqacsIT2H8ijzrtr9wclGZ/KURYs2eAmakdgiiTqjlqN2x1TiauU81vk6/S/bst8ayI3XZY3EX8CsA2ZvJaZ5byXyhXox9d/VdlTLf3k8cZ/CY=" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=1,\\:\\mathrm{Horizontal}:\\:y=1" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x^{2}+4x+3}{x^{2}-1}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }