0}$.","points":null},{"id":"7c2db852-5137-435f-9cfc-3181ef2149ab","cardType":"basic","front":"What happens to the sign of a negative value like $x = -3$ when it is squared vs cubed?","back":"

$(-3)^2$ becomes positive (9)
$(-3)^3$ stays negative (-27)

","points":2},{"id":"ee636513-3474-4c83-99d7-c7c410b64d29","cardType":"cloze","front":"According to the factor theorem, $(x - a)$ is a factor of $f(x)$ if and only if $\\text{[...]}$.","back":"According to the factor theorem, $(x - a)$ is a factor of $f(x)$ if and only if $\\boxed{f(a) = 0}$.","points":null},{"id":"e5e5e739-742b-416f-b8ea-17e2ed7e2d47","cardType":"basic","front":"To test if $(2x + 3)$ is a factor of a polynomial, what value of $x$ should be substituted?","back":"$$x = -\\frac{3}{2}$","points":1},{"id":"44a37ebd-7d3d-4131-8ed7-7b4952002ee8","cardType":"basic","front":"If $f(x) = (x - 2)(Ax^2 + Bx + C)$, how are $A$, $B$, and $C$ typically found at AS level?","back":"Comparing coefficients after expanding the right-hand side.","points":1},{"id":"88e98304-ca6a-4bd8-a266-7d2c33703094","cardType":"basic","front":"Where should you look for possible rational roots of an integer cubic polynomial?","back":"Factors of the constant term divided by factors of the leading coefficient.","points":1},{"id":"194a2eb2-abe7-4393-b6e1-dfe8f90b3355","cardType":"cloze","front":"If a quadratic $ax^2 + bx + c$ has no real roots, its discriminant $\\text{[...]}$ must be $< 0$.","back":"If a quadratic $ax^2 + bx + c$ has no real roots, its discriminant $\\boxed{b^2 - 4ac}$ must be $< 0$.","points":null},{"id":"a3a7ab71-a5fd-4e89-94a7-cf2f32bce031","cardType":"cloze","front":"If a quadratic $ax^2 + bx + c$ has no real roots, its discriminant $b^2 - 4ac$ must be $\\text{[...]}$.","back":"If a quadratic $ax^2 + bx + c$ has no real roots, its discriminant $b^2 - 4ac$ must be $\\boxed{< 0}$.","points":null},{"id":"40098186-8394-42ac-a5b5-f837431880c5","cardType":"basic","front":"How does a repeated root appear on a cubic graph?","back":"The graph touches the $x$-axis and turns around (does not cross).","points":1},{"id":"d3a1666a-f0e6-454c-afa3-fa8d94fe370a","cardType":"basic","front":"How do you show a cubic equation has exactly one real root?","back":"

Show one linear factor $(x - a)$ gives $f(a) = 0$
Show the remaining quadratic factor has a negative discriminant

","points":2},{"id":"fd628800-b2f9-42cc-a486-71557c502eb4","cardType":"cloze","front":"In algebraic proofs, an even integer is written as $\\text{[...]}$ and an odd integer as $2n + 1$.","back":"In algebraic proofs, an even integer is written as $\\boxed{2n}$ and an odd integer as $2n + 1$.","points":null},{"id":"d5333c93-cc18-4ba8-8d36-07ce07bc35b0","cardType":"cloze","front":"In algebraic proofs, an even integer is written as $2n$ and an odd integer as $\\text{[...]}$.","back":"In algebraic proofs, an even integer is written as $2n$ and an odd integer as $\\boxed{2n + 1}$.","points":null},{"id":"18cefa81-b18f-4309-bb68-47714f555ee2","cardType":"basic","front":"How do you disprove the claim: \"$n^2 - n + 3$ is prime for every positive integer $n$\"?","back":"Find a counterexample (e.g., if $n=3$, $3^2 - 3 + 3 = 9$, which is not prime).","points":2},{"id":"a74139e2-22ea-48d0-b7ff-de2d9aec19dd","cardType":"basic","front":"Prove that the sum of two neighbouring odd integers is a multiple of 4.","back":"

Let integers be $2n + 1$ and $2n + 3$
Sum: $(2n + 1) + (2n + 3) = 4n + 4$
Factorise: $4(n + 1)$, which is a multiple of 4

","points":3},{"id":"369834d4-679c-4f63-97dc-3b901fda102e","cardType":"basic","front":"When proving a quadratic inequality is positive for all real $x$, what technique is most effective?","back":"Completing the square.","points":1},{"id":"25198694-3e5e-4fe2-b893-56360f7bdf58","ca

Algebraic Methods Flashcards

Algebraic Methods Flashcards

Algebraic Methods Flashcards

Algebraic Methods Flashcards