AP EAMCET Algebra: Theory of Equations — practice questions
78 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice AP EAMCET Algebra: Theory of Equations in the app →A degree n complex polynomial has how many roots, counted with multiplicity?The fundamental theorem of algebra shows the complex numbers form which kind of field?A real polynomial of odd degree must have at least how many real roots?If 2 + 3i is a root of a real-coefficient polynomial, which is also a root?A real quadratic (monic) has the root 2 + 3i. The quadratic isA real cubic has roots 2 and 1 + 2i. What is its third root?A real polynomial of degree 6 has 4 non-real roots. How many real roots (with multiplicity)?x^4 - 1 = 0 has how many non-real complex roots?In (x - 2)^3 (x + 1)^2, what is the multiplicity of the root 2?Vieta's formulas relate polynomial coefficients to what about its roots?Vieta's formulas are named after which mathematician?Which expression gives the sum of roots of ax^2 + bx + c = 0?Two numbers solve 2x^2 - 9x + 4 = 0. Their total isMultiplying both solutions of 3x^2 + 5x - 12 = 0 givesA quadratic x^2 - 8x + k has 3 as a root. Find k.Which quadratic has roots 2 and -3?With a and b solving x^2 - 4x + 1 = 0, a^2 + b^2 equalsRoots of x^2 - 5x + 6 = 0 are reciprocated and added. The result isGiven x^2 - x - 1 = 0 with roots a, b, evaluate the cube sum a^3 + b^3.Roots of x^2 - 9x + k = 0 are in ratio 1 to 2. Determine k.Adding all three roots of x^3 + 5x^2 - 2x + 8 = 0 givesMultiplying all roots of x^3 + 2x^2 - 5x - 6 = 0 together givesTaking the roots of x^3 - 4x^2 + 7x - 2 = 0 two at a time, their products totalA monic cubic has roots 1, 2 and 3. Its expansion isIf the three roots of x^3 - 9x^2 + 23x - 15 = 0 are in arithmetic progression, the middle one isFor x^3 - 6x^2 + 11x - 6 = 0, adding 1/r over all three roots r givesHow many distinct real roots does a quadratic with zero discriminant have?The expression b^2 - 4ac in the quadratic formula is called theA real quadratic with a negative discriminant hasThe roots of 2x^2 + 3x + 5 = 0 are best described asFor what k does x^2 + kx + 9 = 0 have equal roots?x^2 - 2x + k = 0 has real roots whenSolving x^2 - 6x + 9 = 0 yields how many separate real values of x?The curve y = ax^2 + bx + c with a not zero is aA parabola tangent to the x-axis corresponds toIf a parabola does not meet the x-axis, its roots areA quadratic has real roots 1 and 5. Its axis of symmetry is x =Which Indian mathematician gave an explicit quadratic solution in 628 AD?Which 9th century scholar gave a full solution of the general quadratic?Babylonian mathematicians solved quadratic-type problems as early asWhich famous ratio is the positive root of x^2 - x - 1 = 0?Completing the square leads to which general formula?Descartes's rule of signs counts roots by examining what?Descartes's rule limits positive roots of 3x^3 - 2x^2 + x - 5 = 0 to at mostThe number of positive real roots of x^3 - 1 = 0 is exactlyPositive roots equal the sign changes, or fewer than it byDescartes described his rule of signs in which work?The rational root theorem concerns which kind of roots?For 2x^3 - 5x^2 + x + 6 = 0, which cannot be a rational root?Which is a root of x^3 - 4x^2 + x + 6 = 0?With leading coefficient 1, the rational root theorem becomes theIn a rational root p/q in lowest terms, p must divide theAfter finding a rational root r, divide the polynomial byEvery real polynomial factors into real factors of degreeWhich book was the first textbook to contain a proof of the fundamental theorem?Who made an incomplete first attempt at proving the theorem in 1746?Circles and other conic sections are described by which type of equation?Find the product of the roots of 2x^2 + 7x + 3 = 0.For x^3 - 6x^2 + 11x - 6 = 0, what is the sum of the roots?For x^3 - 7x + 6 = 0, the sum of products of roots taken two at a time isIf the roots of x^2 - 3x + k = 0 have product 4, what is k?The cubic with roots 1, 2 and 3 has x^2 coefficient (monic)If a and b are roots of x^2 - 6x + 4 = 0, then 1/a + 1/b equalsThe quadratic with roots 1 + sqrt(2) and 1 - sqrt(2) isIn x^4 - 10x^3 + 35x^2 - 50x + 24 = 0, the sum of the roots isVieta originally derived his root-coefficient relations for roots that wereWho is credited with first grasping the general root-coefficient principle, beyond positive roots?How many sign changes does x^3 - 3x^2 + x - 5 have in its coefficients?x^3 + 2x^2 + 3x + 4 = 0 has at most how many positive roots?A polynomial has exactly one sign change. How many positive roots does it have?For x^3 + x^2 - x - 1, replacing x by -x gives how many sign changes?In Descartes's rule, a root of multiplicity k is counted asHow many real roots does x^3 - 1 = 0 have?The cubic x^3 - 6x^2 + 11x - 6 = 0 has roots 1, 2 and 3. What is the sum of the squares of its roots?The roots of x^4 - 10x^3 + 35x^2 - 50x + 24 = 0 are 1, 2, 3, 4. Find the sum of pairwise products.In x^2 - 8x + k = 0 one root is three times the other. What is the value of k?What is the sum of the roots of the cubic equation 3x^3 + 12x^2 - 5x + 1 = 0?If a and b are the roots of x^2 - 5x + 6 = 0, what is the value of a^2 + b^2?