Properties of inequalities — addition?
Inequality remains the same when adding/subtracting the same number.
Properties of inequalities — multiplication?
Same sign: inequality stays; negative sign: inequality reverses.
Reciprocal inequality rule — when?
Reverses only if both sides have same sign.
Squaring inequalities — caution?
Direction may change unless signs are known.
Quadratic inequality — highest power?
Degree 2 polynomial inequality.
Sign analysis — factors?
Determine positivity/negativity by factor signs.
Graphical method — parabola?
Use parabola to find where inequality holds.
Critical points — significance?
Where quadratic crosses or touches x-axis.
Rational inequality — critical values?
Values where denominator is zero or numerator zero.
Excluded denominator values — why?
Cannot evaluate; marked with open circles.
Factorised sign test — approach?
Rewrite as product; analyze sign of each factor.
Absolute value inequality — graph?
V-shape, non-negative distance from inside expression.
Square root inequality — domain?
Inside must be ≥ 0 for the expression to exist.
Circle inequality — form?
x² + y² ≤ r² or ≥ r².
Inside circle region — inequality?
x² + y² ≤ r² (including boundary).
Region between curves — condition?
On/between both curves, satisfying all inequalities.
Degree of a polynomial — definition?
Highest power of x with non-zero coefficient.
Leading coefficient — role?
Coefficient of highest-degree term.
Monic polynomial — what?
Leading coefficient equals 1.
Roots of polynomial — what?
Values where P(x)=0.
Polynomial division — key rule?
Dividend = Divisor × Quotient + Remainder.
Remainder theorem — statement?
P(a)=0 iff (x−a) divides P(x) exactly.
Roots and coefficients — relation?
Sum/product linked to coefficients; quadratic: sum = -b/a.
Multiple roots — indicator?
Root also zero of derivative; multiplicity >1.
Teste tes connaissances avec un QCM de 24 questions sur Mastering Polynomial Inequalities and Graphs.
1. What happens to the direction of an inequality when the same real number is added to both sides?
2. If both sides of an inequality are multiplied by a negative number, what must happen to the inequality sign?
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