Questions on Algebra: Polynomials, rational expressions and equations answered by real tutors!

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Question 1209715: Let P(x) be a polynomial of the form
P(x) = 2x^3 + ax^2 - 23x + c,
such that 12 and 7 are roots of P(x). What is the third root?
For the polynomial in part (a), compute the ordered pair (a,c).
Click here to see answer by CPhill(1959)

Question 1209717: Let f(x) be a polynomial. Find the remainder when f(x) is divided by x(x - 1)(x - 2), if f(0) = 0, f(1) = 1, and f(2) = 2.
Click here to see answer by CPhill(1959)

Question 1209720: Suppose p(x) is a monic cubic polynomial with real coefficients such that p(2 - 3i) = 8 and p(0) = -5 and p(4 + 7i) = 11.

Determine p(x) (in expanded form).

Click here to see answer by CPhill(1959)

Question 1209721: For parts (a)-(d), let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.

Compute pqr + pqs + prs + qrs.

Click here to see answer by math_tutor2020(3816)

Question 1209722: Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{1}{s}.
Click here to see answer by math_tutor2020(3816)

Question 1209724: Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute p^2 qrs + pq^2 rs + pqr^2 s + pqrs^2.
Click here to see answer by ikleyn(52777)

Question 1209723: Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute p^2 + q^2 + r^2 + s^2.
Click here to see answer by ikleyn(52777)

Question 1209731: Let r, s, and t be solutions of the equation .
Compute

Click here to see answer by CPhill(1959)

Question 1209730: Let r, s, and t be solutions of the equation x^3 - 4x^2 - 7x + 12 = 0. Compute
\frac{rs}{t^2} + \frac{rt}{s^2} + \frac{st}{r^2}
Click here to see answer by CPhill(1959)

Question 1209729: Let t be a root of f(x) = x^3 - x + 3. Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.

Click here to see answer by CPhill(1959)

Question 1209729: Let t be a root of f(x) = x^3 - x + 3. Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.

Click here to see answer by ikleyn(52777)

Question 1209732: Suppose r and s are the values of x that satisfy the equation
x^2 - 2mx + (m^2 - 6m + 11) = 0
for some real number m. Find the minimum real value of (r - s)^2.
Click here to see answer by ikleyn(52777)

Question 1209738: Let r, s, and t be solutions of the equation 3x^3 - 4x^2 - 2x + 12 = 0. Compute
\frac{rs}{t^2} + \frac{rt}{s^2} + \frac{st}{r^2}.
Click here to see answer by CPhill(1959)

Question 1209739: Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 5n - mn + 20m + 4n + ___
Click here to see answer by CPhill(1959)

Question 1209740: Factor x^2 - 2x - y^2 + 2yz + 5z^2 as the product of two polynomials of degree 1.
Click here to see answer by CPhill(1959)

Question 1209745: Fill in the blanks, to complete the factorization:

(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___)
Click here to see answer by CPhill(1959)

Question 1209746: Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)
Click here to see answer by CPhill(1959)

Question 1209749: If (x,y) satisfies the simultaneous equations
3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20 = 0.
x^2 - y^2 = 7 + 4xy
where x and y may be complex numbers, determine all possible values of y^2.
Click here to see answer by CPhill(1959)

Question 1209748: Factor 3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.
Click here to see answer by CPhill(1959)

Question 1209751: In this multi-part problem, we will consider this system of simultaneous equations:
3x + 4y + 30z = -60,
2xy + 42xz - 16yz = 68,
5xyz = 56.
Let a = x/2, b = 5y and c = -4z.
Determine the monic cubic polynomial in terms of a variable t whose roots are t = a, t = b, and t = c.

[i]This is a continuation of the problem above.[/i]
Given that $(x,y,z)$ is a solution to the original system of equations, determine all distinct possible values of $x + y + z$.
(Suggestion: Using the substitutions in part (a), first determine all possible values of the ordered triple $(a,b,c)$, then determine the possible solutions $(x,y,z)$.)
Click here to see answer by CPhill(1959)

Question 1209757: If z is a complex number satisfying
z + \frac{1}{z} = \sqrt{2},
calculate
z^{10} + \frac{1}{z^{10}}.
Click here to see answer by CPhill(1959)

Question 1209760: Let
p(a, b, c) = a^5 + b^5 + c^5 + k(a^3 + b^3 + c^3)(a^4 + b^4 + c^4 - 2a^2 b^2 - 2a^2 c^2 - 2b^2 c^2).
Find the constant k so that p(a,b,c) is divisible by a + b + c.
Click here to see answer by CPhill(1959)

Question 1209763: Find the constant k so that
2x^2 + 5xy - 8y^2 + 7x + 25y + k
can be expressed as the product of two linear factors of the form ax + by + c.

Click here to see answer by CPhill(1959)

Question 1209764: Express x^8 + x^4 y^4 - x^6 y^2 + x^3 y^5 - 4xy^7 - y^8 as the product of two polynomials of degree 2, and a polynomial of degree 4.
Click here to see answer by CPhill(1959)

Question 1209770: Find all (real or nonreal) x satisfying
(x - 3)^4 + (x - 5)^4 = -8 + 6(x - 3)(x - 5)^3 - 11(x - 3)^3 (x - 5).
Click here to see answer by CPhill(1959)

Question 1209770: Find all (real or nonreal) x satisfying
(x - 3)^4 + (x - 5)^4 = -8 + 6(x - 3)(x - 5)^3 - 11(x - 3)^3 (x - 5).
Click here to see answer by ikleyn(52777)

Question 1209771: Suppose the polynomial p(x)=x^3+ax^2+bc+c has the property that the mean of its zeroes, the product of its zeroes, and the sum of its coefficients are all equal. If the y-intercept of the graph of y=p(x) is 0, what is b?
Click here to see answer by CPhill(1959)

Question 1209772: Let a = 3(x - y), b = 3(y - z), and c = 3(z - x), where x, y, z are real numbers, and assume ab + ac + bc \ne 0. Compute
(a^3 + b^3 + c^3)/(ab + ac + bc).
Click here to see answer by CPhill(1959)

Question 1209789: Draw the graph of y=x²+x+2
For -4≤x≤4

Click here to see answer by CPhill(1959)

Question 1209796: Find all real numbers a such that the roots of the polynomial
x^3 - 3x^2 + 17x + a
form an arithmetic progression and are not all real.
Click here to see answer by greenestamps(13198)

Question 1209800: Simplify (x^4 + x^3 + x^2 + x + 1) + (x^4 - x^3 + x^2 - x + 1).
Click here to see answer by CPhill(1959)

Question 1209800: Simplify (x^4 + x^3 + x^2 + x + 1) + (x^4 - x^3 + x^2 - x + 1).
Click here to see answer by Edwin McCravy(20054)

Question 1209802: Let A = x^4 + x^3 + x^2 + x + 1 and B = x^4 - x^3 + x^2 - x + 1. Simplify A + B.
Click here to see answer by CPhill(1959)

Question 1209802: Let A = x^4 + x^3 + x^2 + x + 1 and B = x^4 - x^3 + x^2 - x + 1. Simplify A + B.
Click here to see answer by greenestamps(13198)

Question 1174083: Can someone please help me with this math riddle? I have been struggling with it for days now.
Can you help me think of a Polynomial (With using ax^2+bx+c) where a Polynomial of b, P(b) is divisible by a-b, and have the quotient of this be composite? I have tried so many different values for the a,b, and c in this polynomial, and I just don't know where to go off of here. Thank you so much for your time and help!
Click here to see answer by CPhill(1959)

Question 1209838: Factor (ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3 + (a^2 + b^2 + c^2)^3 - 3(a^3 + b^3 + c^3)^2 as much as possible.
Click here to see answer by CPhill(1959)

Question 1171540: What can be said about the domain of the function f . g where f(y)= {4}/{y-2} and g(x)= {5}/{3x-1} ? Express it in terms of a union of intervals of real numbers. Go to www.desmos.com/calculator and obtain the graph of f , g , and f. g .
Find the inverse of the function f(x)=4+ \sqrt{x-2} .
State the domains and ranges of both the function and the inverse function in terms of intervals of real numbers.
Go to www.desmos.com/calculator and obtain the graph of f , its inverse, and g(x)=x in the same system of axes. About what pair (a, a) are (11, 7) and (7, 11) reflected about?
Click here to see answer by CPhill(1959)

Question 1171467: What can be said about the domain of the function f \circ g where f(y)= \frac{4}{y-2} and g(x)= \frac{5}{3x-1} ? Express it in terms of a union of intervals of real numbers. Go to www.desmos.com/calculator and obtain the graph of f , g , and f \circ g .
Find the inverse of the function f(x)=4+ \sqrt{x-2} .
State the domains and ranges of both the function and the inverse function in terms of intervals of real numbers.
Go to www.desmos.com/calculator and obtain the graph of f , its inverse, and g(x)=x in the same system of axes. About what pair (a, a) are (11, 7) and (7, 11) reflected about?
Click here to see answer by CPhill(1959)

Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645, 3646..3690, 3691..3735, 3736..3780, 3781..3825, 3826..3870, 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7606..7650, 7651..7695, 7696..7740, 7741..7785, 7786..7830, 7831..7875, 7876..7920, 7921..7965, 7966..8010, 8011..8055, 8056..8100, 8101..8145, 8146..8190, 8191..8235, 8236..8280, 8281..8325, 8326..8370, 8371..8415, 8416..8460, 8461..8505, 8506..8550, 8551..8595, 8596..8640, 8641..8685, 8686..8730, 8731..8775, 8776..8820, 8821..8865, 8866..8910, 8911..8955, 8956..9000, 9001..9045, 9046..9090, 9091..9135, 9136..9180, 9181..9225, 9226..9270, 9271..9315, 9316..9360, 9361..9405, 9406..9450, 9451..9495, 9496..9540, 9541..9585, 9586..9630, 9631..9675, 9676..9720, 9721..9765, 9766..9810, 9811..9855, 9856..9900, 9901..9945, 9946..9990, 9991..10035, 10036..10080, 10081..10125, 10126..10170, 10171..10215, 10216..10260, 10261..10305, 10306..10350, 10351..10395, 10396..10440, 10441..10485, 10486..10530, 10531..10575, 10576..10620, 10621..10665, 10666..10710, 10711..10755, 10756..10800, 10801..10845, 10846..10890, 10891..10935, 10936..10980, 10981..11025, 11026..11070, 11071..11115, 11116..11160, 11161..11205, 11206..11250, 11251..11295, 11296..11340, 11341..11385, 11386..11430, 11431..11475, 11476..11520, 11521..11565, 11566..11610, 11611..11655, 11656..11700, 11701..11745, 11746..11790, 11791..11835, 11836..11880, 11881..11925, 11926..11970, 11971..12015, 12016..12060, 12061..12105, 12106..12150, 12151..12195, 12196..12240, 12241..12285, 12286..12330, 12331..12375, 12376..12420, 12421..12465, 12466..12510, 12511..12555, 12556..12600, 12601..12645, 12646..12690, 12691..12735, 12736..12780, 12781..12825, 12826..12870, 12871..12915, 12916..12960, 12961..13005, 13006..13050, 13051..13095, 13096..13140, 13141..13185, 13186..13230, 13231..13275, 13276..13320, 13321..13365, 13366..13410, 13411..13455, 13456..13500, 13501..13545, 13546..13590, 13591..13635, 13636..13680, 13681..13725, 13726..13770, 13771..13815, 13816..13860, 13861..13905, 13906..13950, 13951..13995, 13996..14040, 14041..14085, 14086..14130, 14131..14175, 14176..14220, 14221..14265, 14266..14310, 14311..14355, 14356..14400, 14401..14445, 14446..14490, 14491..14535, 14536..14580, 14581..14625, 14626..14670, 14671..14715, 14716..14760, 14761..14805, 14806..14850, 14851..14895, 14896..14940, 14941..14985, 14986..15030, 15031..15075, 15076..15120, 15121..15165, 15166..15210, 15211..15255, 15256..15300, 15301..15345, 15346..15390, 15391..15435, 15436..15480, 15481..15525, 15526..15570, 15571..15615, 15616..15660, 15661..15705, 15706..15750, 15751..15795, 15796..15840, 15841..15885, 15886..15930, 15931..15975, 15976..16020, 16021..16065, 16066..16110, 16111..16155, 16156..16200, 16201..16245, 16246..16290, 16291..16335, 16336..16380, 16381..16425, 16426..16470, 16471..16515, 16516..16560, 16561..16605, 16606..16650, 16651..16695, 16696..16740, 16741..16785, 16786..16830, 16831..16875, 16876..16920, 16921..16965, 16966..17010, 17011..17055, 17056..17100, 17101..17145, 17146..17190, 17191..17235, 17236..17280, 17281..17325, 17326..17370, 17371..17415, 17416..17460, 17461..17505, 17506..17550, 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20746..20790, 20791..20835, 20836..20880, 20881..20925, 20926..20970, 20971..21015, 21016..21060, 21061..21105, 21106..21150, 21151..21195, 21196..21240, 21241..21285, 21286..21330, 21331..21375, 21376..21420, 21421..21465, 21466..21510, 21511..21555, 21556..21600, 21601..21645, 21646..21690, 21691..21735, 21736..21780, 21781..21825, 21826..21870, 21871..21915, 21916..21960, 21961..22005, 22006..22050, 22051..22095, 22096..22140, 22141..22185, 22186..22230, 22231..22275, 22276..22320, 22321..22365, 22366..22410, 22411..22455, 22456..22500, 22501..22545, 22546..22590, 22591..22635, 22636..22680, 22681..22725, 22726..22770, 22771..22815, 22816..22860, 22861..22905, 22906..22950, 22951..22995, 22996..23040, 23041..23085, 23086..23130, 23131..23175, 23176..23220, 23221..23265, 23266..23310, 23311..23355, 23356..23400, 23401..23445, 23446..23490, 23491..23535, 23536..23580, 23581..23625, 23626..23670, 23671..23715, 23716..23760, 23761..23805, 23806..23850, 23851..23895, 23896..23940, 23941..23985, 23986..24030, 24031..24075, 24076..24120, 24121..24165, 24166..24210, 24211..24255, 24256..24300, 24301..24345, 24346..24390, 24391..24435, 24436..24480, 24481..24525, 24526..24570, 24571..24615, 24616..24660, 24661..24705, 24706..24750, 24751..24795, 24796..24840, 24841..24885, 24886..24930, 24931..24975, 24976..25020, 25021..25065, 25066..25110, 25111..25155, 25156..25200, 25201..25245, 25246..25290, 25291..25335, 25336..25380, 25381..25425, 25426..25470, 25471..25515, 25516..25560, 25561..25605, 25606..25650, 25651..25695, 25696..25740, 25741..25785, 25786..25830, 25831..25875, 25876..25920, 25921..25965, 25966..26010, 26011..26055, 26056..26100, 26101..26145, 26146..26190, 26191..26235, 26236..26280, 26281..26325, 26326..26370, 26371..26415, 26416..26460, 26461..26505, 26506..26550, 26551..26595, 26596..26640, 26641..26685, 26686..26730, 26731..26775, 26776..26820, 26821..26865, 26866..26910, 26911..26955, 26956..27000, 27001..27045, 27046..27090, 27091..27135, 27136..27180, 27181..27225, 27226..27270, 27271..27315, 27316..27360, 27361..27405, 27406..27450, 27451..27495, 27496..27540, 27541..27585, 27586..27630, 27631..27675, 27676..27720, 27721..27765, 27766..27810, 27811..27855, 27856..27900, 27901..27945, 27946..27990, 27991..28035, 28036..28080, 28081..28125, 28126..28170, 28171..28215, 28216..28260, 28261..28305, 28306..28350, 28351..28395, 28396..28440, 28441..28485, 28486..28530, 28531..28575, 28576..28620, 28621..28665, 28666..28710, 28711..28755, 28756..28800, 28801..28845, 28846..28890, 28891..28935, 28936..28980, 28981..29025, 29026..29070, 29071..29115, 29116..29160, 29161..29205, 29206..29250, 29251..29295, 29296..29340, 29341..29385, 29386..29430, 29431..29475, 29476..29520, 29521..29565, 29566..29610, 29611..29655, 29656..29700, 29701..29745, 29746..29790, 29791..29835, 29836..29880, 29881..29925, 29926..29970, 29971..30015, 30016..30060, 30061..30105, 30106..30150, 30151..30195, 30196..30240, 30241..30285, 30286..30330, 30331..30375, 30376..30420, 30421..30465, 30466..30510, 30511..30555, 30556..30600, 30601..30645, 30646..30690, 30691..30735, 30736..30780, 30781..30825, 30826..30870, 30871..30915, 30916..30960, 30961..31005, 31006..31050, 31051..31095, 31096..31140, 31141..31185, 31186..31230, 31231..31275, 31276..31320, 31321..31365, 31366..31410, 31411..31455, 31456..31500, 31501..31545, 31546..31590, 31591..31635, 31636..31680, 31681..31725, 31726..31770, 31771..31815, 31816..31860, 31861..31905, 31906..31950, 31951..31995, 31996..32040, 32041..32085, 32086..32130, 32131..32175, 32176..32220, 32221..32265, 32266..32310, 32311..32355, 32356..32400, 32401..32445, 32446..32490, 32491..32535, 32536..32580, 32581..32625, 32626..32670, 32671..32715, 32716..32760, 32761..32805, 32806..32850, 32851..32895, 32896..32940, 32941..32985, 32986..33030, 33031..33075, 33076..33120, 33121..33165, 33166..33210, 33211..33255, 33256..33300, 33301..33345, 33346..33390, 33391..33435, 33436..33480, 33481..33525, 33526..33570, 33571..33615, 33616..33660, 33661..33705, 33706..33750, 33751..33795, 33796..33840, 33841..33885, 33886..33930, 33931..33975, 33976..34020, 34021..34065, 34066..34110, 34111..34155, 34156..34200, 34201..34245, 34246..34290, 34291..34335, 34336..34380, 34381..34425, 34426..34470, 34471..34515, 34516..34560, 34561..34605, 34606..34650, 34651..34695, 34696..34740, 34741..34785, 34786..34830, 34831..34875, 34876..34920, 34921..34965, 34966..35010, 35011..35055, 35056..35100, 35101..35145, 35146..35190, 35191..35235, 35236..35280, 35281..35325, 35326..35370, 35371..35415, 35416..35460, 35461..35505, 35506..35550, 35551..35595, 35596..35640, 35641..35685, 35686..35730, 35731..35775, 35776..35820, 35821..35865, 35866..35910, 35911..35955, 35956..36000, 36001..36045, 36046..36090, 36091..36135, 36136..36180, 36181..36225, 36226..36270, 36271..36315, 36316..36360, 36361..36405, 36406..36450, 36451..36495, 36496..36540, 36541..36585, 36586..36630, 36631..36675, 36676..36720, 36721..36765, 36766..36810, 36811..36855, 36856..36900, 36901..36945, 36946..36990, 36991..37035, 37036..37080, 37081..37125, 37126..37170, 37171..37215, 37216..37260, 37261..37305, 37306..37350, 37351..37395, 37396..37440, 37441..37485, 37486..37530, 37531..37575, 37576..37620, 37621..37665, 37666..37710, 37711..37755, 37756..37800, 37801..37845, 37846..37890, 37891..37935, 37936..37980, 37981..38025, 38026..38070, 38071..38115, 38116..38160, 38161..38205, 38206..38250, 38251..38295, 38296..38340, 38341..38385, 38386..38430, 38431..38475, 38476..38520, 38521..38565, 38566..38610, 38611..38655, 38656..38700, 38701..38745, 38746..38790, 38791..38835, 38836..38880, 38881..38925, 38926..38970, 38971..39015, 39016..39060, 39061..39105