factor 4x^3-13x^2+4x-3

{ "query": { "display": "factor $$4x^{3}-13x^{2}+4x-3$$", "symbolab_question": "FACTOR#factor 4x^{3}-13x^{2}+4x-3" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "(x-3)(4x^{2}-x+1)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$4x^{3}-13x^{2}+4x-3:{\\quad}\\left(x-3\\right)\\left(4x^{2}-x+1\\right)$$", "input": "4x^{3}-13x^{2}+4x-3", "steps": [ { "type": "interim", "title": "Use the rational root theorem", "input": "4x^{3}-13x^{2}+4x-3", "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": [ "$$a_{0}=3,\\:{\\quad}a_{n}=4$$" ] }, { "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": "Factors of $$4:{\\quad}1,\\:2,\\:4$$", "input": "4", "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 $$4:{\\quad}2,\\:2$$", "input": "4", "steps": [ { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2" }, { "type": "step", "primary": "$$2$$ is a prime number, therefore no further factorization is possible", "result": "=2\\cdot\\:2" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRrhUN1mnSHVX3em1snAD0TLwt9LEn7QCBUukJKctfSJKeJuqQqR+l8XRZrIWP4OvJnpvpbUsUKOzyFoiTFrkVr6/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2" }, { "type": "step", "primary": "Add 1 and the number $$4\\:$$ itself", "result": "1,\\:4" }, { "type": "step", "primary": "The factors of $$4$$", "result": "1,\\:2,\\:4" } ], "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: $$4x^{3}-13x^{2}+4x-3$$", "steps": [ { "type": "step", "primary": "Plug each combination of $$\\frac{\\pm\\:u}{v}$$ into $$4x^{3}-13x^{2}+4x-3\\:$$ until a root is found:", "secondary": [ "Plug $$\\frac{1}{1}\\:$$into $$4x^{3}-13x^{2}+4x-3\\:$$and check if result is 0: False", "Plug $$\\frac{-1}{1}\\:$$into $$4x^{3}-13x^{2}+4x-3\\:$$and check if result is 0: False", "Plug $$\\frac{1}{2}\\:$$into $$4x^{3}-13x^{2}+4x-3\\:$$and check if result is 0: False", "Plug $$\\frac{-1}{2}\\:$$into $$4x^{3}-13x^{2}+4x-3\\:$$and check if result is 0: False", "Plug $$\\frac{1}{4}\\:$$into $$4x^{3}-13x^{2}+4x-3\\:$$and check if result is 0: False", "Plug $$\\frac{-1}{4}\\:$$into $$4x^{3}-13x^{2}+4x-3\\:$$and check if result is 0: False", "Plug $$\\frac{3}{1}\\:$$into $$4x^{3}-13x^{2}+4x-3\\:$$and check if result is 0: True" ] } ], "meta": { "interimType": "Rational Root Check Title 1Eq" } }, { "type": "step", "primary": "$$u=3,\\:v=1$$" }, { "type": "step", "primary": "Multiply polynom by $$\\frac{v*x-u}{v*x-u}$$", "secondary": [ "$$\\frac{v*x-u}{v*x-u}=\\frac{1\\cdot\\:x-3}{1\\cdot\\:x-3}=\\frac{x-3}{x-3}$$" ], "result": "=\\left(x-3\\right)\\frac{4x^{3}-13x^{2}+4x-3}{x-3}" } ], "meta": { "interimType": "Rational Root Theorem Title 0Eq" } }, { "type": "step", "result": "=\\left(x-3\\right)\\frac{4x^{3}-13x^{2}+4x-3}{x-3}" }, { "type": "interim", "title": "$$\\frac{4x^{3}-13x^{2}+4x-3}{x-3}=4x^{2}-x+1$$", "input": "\\frac{4x^{3}-13x^{2}+4x-3}{x-3}", "steps": [ { "type": "interim", "title": "Divide $$\\frac{4x^{3}-13x^{2}+4x-3}{x-3}:{\\quad}\\frac{4x^{3}-13x^{2}+4x-3}{x-3}=4x^{2}+\\frac{-x^{2}+4x-3}{x-3}$$", "result": "=4x^{2}+\\frac{-x^{2}+4x-3}{x-3}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$4x^{3}-13x^{2}+4x-3$$<br/>and the divisor $$x-3\\::\\:\\frac{4x^{3}}{x}=4x^{2}$$", "result": "\\mathrm{Quotient}=4x^{2}" }, { "type": "step", "primary": "Multiply $$x-3$$ by $$4x^{2}:\\:4x^{3}-12x^{2}$$", "secondary": [ "Subtract $$4x^{3}-12x^{2}$$ from $$4x^{3}-13x^{2}+4x-3$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-x^{2}+4x-3" }, { "type": "step", "primary": "Therefore", "result": "\\frac{4x^{3}-13x^{2}+4x-3}{x-3}=4x^{2}+\\frac{-x^{2}+4x-3}{x-3}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{-x^{2}+4x-3}{x-3}:{\\quad}\\frac{-x^{2}+4x-3}{x-3}=-x+\\frac{x-3}{x-3}$$", "result": "=4x^{2}-x+\\frac{x-3}{x-3}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$-x^{2}+4x-3$$<br/>and the divisor $$x-3\\::\\:\\frac{-x^{2}}{x}=-x$$", "result": "\\mathrm{Quotient}=-x" }, { "type": "step", "primary": "Multiply $$x-3$$ by $$-x:\\:-x^{2}+3x$$", "secondary": [ "Subtract $$-x^{2}+3x$$ from $$-x^{2}+4x-3$$ to get new remainder" ], "result": "\\mathrm{Remainder}=x-3" }, { "type": "step", "primary": "Therefore", "result": "\\frac{-x^{2}+4x-3}{x-3}=-x+\\frac{x-3}{x-3}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{x-3}{x-3}:{\\quad}\\frac{x-3}{x-3}=1$$", "result": "=4x^{2}-x+1", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$x-3$$<br/>and the divisor $$x-3\\::\\:\\frac{x}{x}=1$$", "result": "\\mathrm{Quotient}=1" }, { "type": "step", "primary": "Multiply $$x-3$$ by $$1:\\:x-3$$", "secondary": [ "Subtract $$x-3$$ from $$x-3$$ to get new remainder" ], "result": "\\mathrm{Remainder}=0" }, { "type": "step", "primary": "Therefore", "result": "\\frac{x-3}{x-3}=1" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\left(x-3\\right)\\left(4x^{2}-x+1\\right)" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }