Flashcard: Mastering Polynomial Inequalities and Graphs — 24 carte

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.