factor 8x^3-62x^2+43x-7

{ "query": { "display": "factor $$8x^{3}-62x^{2}+43x-7$$", "symbolab_question": "FACTOR#factor 8x^{3}-62x^{2}+43x-7" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "(2x-1)(4x-1)(x-7)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$8x^{3}-62x^{2}+43x-7:{\\quad}\\left(2x-1\\right)\\left(4x-1\\right)\\left(x-7\\right)$$", "input": "8x^{3}-62x^{2}+43x-7", "steps": [ { "type": "interim", "title": "Use the rational root theorem", "input": "8x^{3}-62x^{2}+43x-7", "steps": [ { "type": "definition", "title": "Rational root theorem definition", "text": "For a polynomial equation with integer coefficients:$${\\quad}a_{n}x^{n}+a_{n-1}x^{n-1}+\\ldots+a_{0}$$<br/>If $$a_{0}$$ and $$a_{n}$$ are integers, then if there is a rational solution<br/>it could be found by checking all the numbers produced for $$\\frac{\\pm\\:\\mathrm{dividers\\:of}\\:a_{0}}{\\mathrm{dividers\\:of}\\:a_{n}}$$", "secondary": [ "$$a_{0}=7,\\:{\\quad}a_{n}=8$$" ] }, { "type": "interim", "title": "Factors of $$7:{\\quad}1,\\:7$$", "input": "7", "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 $$7:{\\quad}7$$", "input": "7", "steps": [ { "type": "step", "primary": "$$7$$ is a prime number, therefore no factorization is possible", "result": "=7" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsuo5jZiW0T+uPgKywRVRozwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJq+H1JIem2SYFK9CRPpgbIy/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add 1 ", "result": "1" }, { "type": "step", "primary": "The factors of $$7$$", "result": "1,\\:7" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Factors of $$8:{\\quad}1,\\:2,\\:4,\\:8$$", "input": "8", "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 $$8:{\\quad}2,\\:2,\\:2$$", "input": "8", "steps": [ { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "$$2$$ is a prime number, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRlrHmICBBU/V4v9mDHayw0Dwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJiL6o6RfUkqI5hqGsUSBhPK/Mg94S0N9we//Py6WzxN6" } }, { "type": "interim", "title": "Multiply the prime factors of $$8:{\\quad}4$$", "result": "4", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$" }, { "type": "step", "result": "4" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2" }, { "type": "step", "primary": "Add 1 and the number $$8\\:$$ itself", "result": "1,\\:8" }, { "type": "step", "primary": "The factors of $$8$$", "result": "1,\\:2,\\:4,\\:8" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "For every $$a_{0}$$ factor $$u$$ and $$a_{n}$$ factor $$v$$, check if $$\\frac{\\pm\\:u}{v}$$ is a root of: $$8x^{3}-62x^{2}+43x-7$$", "steps": [ { "type": "step", "primary": "Plug each combination of $$\\frac{\\pm\\:u}{v}$$ into $$8x^{3}-62x^{2}+43x-7\\:$$ until a root is found:", "secondary": [ "Plug $$\\frac{1}{1}\\:$$into $$8x^{3}-62x^{2}+43x-7\\:$$and check if result is 0: False", "Plug $$\\frac{-1}{1}\\:$$into $$8x^{3}-62x^{2}+43x-7\\:$$and check if result is 0: False", "Plug $$\\frac{1}{2}\\:$$into $$8x^{3}-62x^{2}+43x-7\\:$$and check if result is 0: True" ] } ], "meta": { "interimType": "Rational Root Check Title 1Eq" } }, { "type": "step", "primary": "$$u=1,\\:v=2$$" }, { "type": "step", "primary": "Multiply polynom by $$\\frac{v*x-u}{v*x-u}$$", "secondary": [ "$$\\frac{v*x-u}{v*x-u}=\\frac{2x-1}{2x-1}$$" ], "result": "=\\left(2x-1\\right)\\frac{8x^{3}-62x^{2}+43x-7}{2x-1}" } ], "meta": { "interimType": "Rational Root Theorem Title 0Eq" } }, { "type": "step", "result": "=\\left(2x-1\\right)\\frac{8x^{3}-62x^{2}+43x-7}{2x-1}" }, { "type": "interim", "title": "$$\\frac{8x^{3}-62x^{2}+43x-7}{2x-1}=4x^{2}-29x+7$$", "input": "\\frac{8x^{3}-62x^{2}+43x-7}{2x-1}", "steps": [ { "type": "interim", "title": "Divide $$\\frac{8x^{3}-62x^{2}+43x-7}{2x-1}:{\\quad}\\frac{8x^{3}-62x^{2}+43x-7}{2x-1}=4x^{2}+\\frac{-58x^{2}+43x-7}{2x-1}$$", "result": "=4x^{2}+\\frac{-58x^{2}+43x-7}{2x-1}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$8x^{3}-62x^{2}+43x-7$$<br/>and the divisor $$2x-1\\::\\:\\frac{8x^{3}}{2x}=4x^{2}$$", "result": "\\mathrm{Quotient}=4x^{2}" }, { "type": "step", "primary": "Multiply $$2x-1$$ by $$4x^{2}:\\:8x^{3}-4x^{2}$$", "secondary": [ "Subtract $$8x^{3}-4x^{2}$$ from $$8x^{3}-62x^{2}+43x-7$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-58x^{2}+43x-7" }, { "type": "step", "primary": "Therefore", "result": "\\frac{8x^{3}-62x^{2}+43x-7}{2x-1}=4x^{2}+\\frac{-58x^{2}+43x-7}{2x-1}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{-58x^{2}+43x-7}{2x-1}:{\\quad}\\frac{-58x^{2}+43x-7}{2x-1}=-29x+\\frac{14x-7}{2x-1}$$", "result": "=4x^{2}-29x+\\frac{14x-7}{2x-1}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$-58x^{2}+43x-7$$<br/>and the divisor $$2x-1\\::\\:\\frac{-58x^{2}}{2x}=-29x$$", "result": "\\mathrm{Quotient}=-29x" }, { "type": "step", "primary": "Multiply $$2x-1$$ by $$-29x:\\:-58x^{2}+29x$$", "secondary": [ "Subtract $$-58x^{2}+29x$$ from $$-58x^{2}+43x-7$$ to get new remainder" ], "result": "\\mathrm{Remainder}=14x-7" }, { "type": "step", "primary": "Therefore", "result": "\\frac{-58x^{2}+43x-7}{2x-1}=-29x+\\frac{14x-7}{2x-1}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{14x-7}{2x-1}:{\\quad}\\frac{14x-7}{2x-1}=7$$", "result": "=4x^{2}-29x+7", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$14x-7$$<br/>and the divisor $$2x-1\\::\\:\\frac{14x}{2x}=7$$", "result": "\\mathrm{Quotient}=7" }, { "type": "step", "primary": "Multiply $$2x-1$$ by $$7:\\:14x-7$$", "secondary": [ "Subtract $$14x-7$$ from $$14x-7$$ to get new remainder" ], "result": "\\mathrm{Remainder}=0" }, { "type": "step", "primary": "Therefore", "result": "\\frac{14x-7}{2x-1}=7" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\left(2x-1\\right)\\left(4x^{2}-29x+7\\right)" }, { "type": "interim", "title": "Factor $$4x^{2}-29x+7:{\\quad}\\left(4x-1\\right)\\left(x-7\\right)$$", "input": "4x^{2}-29x+7", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "4x^{2}-29x+7", "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": [ "$$a=4,\\:b=-29,\\:c=7$$", "$$u*v=28,\\:u+v=-29$$" ] }, { "type": "interim", "title": "Factors of $$28:{\\quad}1,\\:2,\\:4,\\:7,\\:14,\\:28$$", "input": "28", "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 $$28:{\\quad}2,\\:2,\\:7$$", "input": "28", "steps": [ { "type": "step", "primary": "$$28\\:$$divides by $$2\\quad\\:28=14\\cdot\\:2$$", "result": "=2\\cdot\\:14" }, { "type": "step", "primary": "$$14\\:$$divides by $$2\\quad\\:14=7\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:7" }, { "type": "step", "primary": "$$2,\\:7$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:7" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRv1shHU2p/O8TrWcbPlb3LmBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKptlmTzO+oYqKr2Nj1dLY7LooWAuAHKtvQokwVYAX2ga" } }, { "type": "interim", "title": "Multiply the prime factors of $$28:{\\quad}4,\\:14$$", "result": "4,\\:14", "steps": [ { "type": "step", "primary": "$$2\\cdot\\:2=4$$", "secondary": [ "$$2\\cdot\\:7=14$$" ] }, { "type": "step", "result": "4,\\:14" } ], "meta": { "interimType": "Multiply the prime factors 1Eq" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:7" }, { "type": "step", "primary": "Add 1 and the number $$28\\:$$ itself", "result": "1,\\:28" }, { "type": "step", "primary": "The factors of $$28$$", "result": "1,\\:2,\\:4,\\:7,\\:14,\\:28" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$28:{\\quad}-1,\\:-2,\\:-4,\\:-7,\\:-14,\\:-28$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-4,\\:-7,\\:-14,\\:-28" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=28,\\:$$check if $$u+v=-29$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=28:\\quad\\:u*v=28,\\:u+v=29\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=2,\\:v=14:\\quad\\:u*v=28,\\:u+v=16\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=4,\\:v=7:\\quad\\:u*v=28,\\:u+v=11\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-1,\\:v=-28:\\quad\\:u*v=28,\\:u+v=-29\\quad\\Rightarrow\\quad\\:$$True", "Check $$u=-2,\\:v=-14:\\quad\\:u*v=28,\\:u+v=-16\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=-4,\\:v=-7:\\quad\\:u*v=28,\\:u+v=-11\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=-1,\\:v=-28" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(4x^{2}-x\\right)+\\left(-28x+7\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwsvwT+JbJqd88PyqtdBtQ2JJQz2lqSQogu9PoWz88zfnRIVQdYp0ZW/Nzw1XCNMUXh4Aty1i0DpNCDInxApaAMNVEJRQ8vkNcydH9zxhOCIRor34XXNQpdY6EjjY0FIXVE3kCh3oevUunZ7/b0qFKBSWsWV28wexhpgboh1m5Hi0Fimz7IJvR01OX8go+hj05ompXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "=\\left(4x^{2}-x\\right)+\\left(-28x+7\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$4x^{2}-x:\\quad\\:x\\left(4x-1\\right)$$", "input": "4x^{2}-x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=4xx-x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(4x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tU3+QUvFDhJv1/CD4ZzMUCL+WyeT/M6UkOYpVjlDEAB1C4k27jzuj6KLnWyn2hb8ZahDJ/HwALQKkKbI9g4ZYhZGW7RUq1NM81GR3HUWqjOF4aOVLaBuuVKS9lqM99wwcP7ZdSik0qQkgZUtBUUfOB0AUqFQbGwAJzmClS9bLrxVTa/ef3q12BQXsl9e9iY5" } }, { "type": "interim", "title": "Factor out $$-7\\:$$from $$-28x+7:\\quad\\:-7\\left(4x-1\\right)$$", "input": "-28x+7", "steps": [ { "type": "step", "primary": "Rewrite $$-28$$ as $$-7\\cdot\\:4$$", "secondary": [ "Rewrite $$7$$ as $$-7\\cdot\\:1$$" ], "result": "=-7\\cdot\\:4x+7\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$-7$$", "result": "=-7\\left(4x-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SwKYqWaIEP4o4W284pl3TByOUtErWNRXbTqqjj69E3uBgn/GSIMdAWH6UHJYQzHaqYgfx/SM6bFWa7aXqdH1NOiGloKKN9A2push0xMbRtt24affICWdjh78fVRVNyyCPHkx8SnG6hao0FB49wn54gG+iDtotTNqpiGfb8VE9kW/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "=x\\left(4x-1\\right)-7\\left(4x-1\\right)" }, { "type": "step", "primary": "Factor out common term $$4x-1$$", "result": "=\\left(4x-1\\right)\\left(x-7\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "=\\left(2x-1\\right)\\left(4x-1\\right)\\left(x-7\\right)" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }