Revision sheet: Mastering Polynomial Inequalities and Graphs

📋 Course Outline

1. Properties of inequalities
2. Quadratic inequalities
3. Rational inequalities
4. Absolute value and square root inequalities
5. Circle and region inequalities
6. Polynomial basics
7. Polynomial operations and division
8. Remainder and factor theorems
9. Roots and coefficients
10. Multiple roots of polynomials
11. Polynomial functions
12. Graphs of polynomial functions

📖 1. Properties of inequalities

🔑 Key Concepts & Definitions

• Adding or subtracting same number : An inequality keeps the same direction when you add or subtract the same real number to both sides.
• Multiplying by positive number : An inequality keeps the same direction when you multiply both sides by a positive real number.
• Multiplying by negative number : An inequality reverses direction when both sides are multiplied by a negative real number.
• Reciprocal inequality rule : Taking reciprocals reverses inequality direction only when both sides have the same sign.
• Squaring inequalities : Squaring both sides can change the inequality direction unless you know the relative size and sign conditions.

📝 Essential Points

• If $a>b$, adding or subtracting the same number to both sides gives an equivalent inequality.
• If you multiply an inequality by a negative number, every strict or non-strict inequality sign reverses direction.
• When both sides are positive or both are negative, from $a>b$ you get $1/a<1/b$ after reciprocals are taken.
• If you square an inequality without sign conditions, the direction may not be preserved because squaring removes sign information.
• For $a,b>0$ with $a>b$, squaring preserves direction: $a^2>b^2$.
• You may take a square root only when both sides are non-negative, and the inequality direction stays the same for $a,b\ge 0$.

💡 Memory Hook

Signs control direction: + keeps, ×(−) flips, reciprocal flips only if both sides share the same sign.

📖 2. Quadratic inequalities

🔑 Key Concepts & Definitions

• Quadratic inequality : A quadratic inequality is an inequality whose highest power of the variable is 2, such as $ax^2+bx+c>0$ or $\ge 0$.
• Sign analysis of factors : Sign analysis determines when a factored product is positive or negative by requiring both factors to have matching signs.
• Graphical method (parabola) : The graphical method solves a quadratic inequality by sketching the parabola and reading the x-values where it lies above or below the x-axis.
• Critical points (roots) : Critical points are the x-values where the quadratic crosses or touches the x-axis, marking where the inequality changes from true to false.

📝 Essential Points

• Factoring like an equation can be wrong for inequalities because the solution set depends on when the product is positive, not just when it equals 0.
• For $x(x-4)>0$, both factors must be positive or both must be negative, giving $x<0$ or $x>4$.
• A parabola $y=x^2-4x$ is concave up and solves $y>0$ by taking x-values where the graph is above the x-axis.
• For $x^2(x-2)(x+1)\ge 0$, a double zero at $x=0$ is included since touching at the x-axis still satisfies $\ge 0$.
• If an inequality uses $\le$ or $\ge$, include the x-values where the parabola cuts or touches the x-axis; use endpoints accordingly.

💡 Memory Hook

“Product positive” rule: $A\cdot B>0$ means $A,B$ share the same sign (both + or both −), which fixes the error from $A=0$ factoring.

📖 3. Rational inequalities

🔑 Key Concepts & Definitions

• Critical values : Critical values are the x values that must be checked or excluded because they make a rational expression undefined or they mark boundary points for the inequality.
• Excluded denominator values : Excluded denominator values are x values that are removed from the solution set because the denominator becomes 0 and the rational expression cannot be evaluated.
• Factorised sign test : Factorised sign test is solving a rational inequality by rewriting it as a product that can be compared to 0 to determine where it is positive or negative.
• Rational inequality boundary : Rational inequality boundaries come from solving the corresponding equation where the rational expression equals the comparison value, creating the cut points for the number line.

📝 Essential Points

• When solving $\frac{1}{x-2}\ge -1$, you first exclude $x=2$ and then solve $\frac{1}{x-2}=-1$ to get the boundary $x=1$.
• If an inequality is multiplied by a squared factor like $(x-2)^2$, the inequality sign does not change because $(x-2)^2$ is always positive (except at $x=2$, which is already excluded).
• For rational inequalities, mark excluded values with open circles and include or exclude boundary solutions based on whether the inequality uses $<$, $>$, $\le$, or $\ge$.
• To avoid case-splitting, rewrite the inequality so the difference is a single rational expression with denominator that you square before multiplying.

💡 Memory Hook

Denominator 0 ⇒ open circle; compare to 0 after making a product via factorisation.

📖 4. Absolute value and square root inequalities

🔑 Key Concepts & Definitions

• Absolute value graph : The absolute value graph forms a V-shape where y equals the non-negative distance from the expression inside the bars.
• Non-negative sides for squaring : When both sides of an inequality are known to be non-negative, squaring can be used without changing the inequality’s direction.
• Semicircle domain : For inequalities involving a square root like $\sqrt{1-x^2}$, the function exists only where the inside is non-negative, giving a restricted x-range.
• Intersection points : Intersection points are x-values where the absolute value graph and the square-root/semicircle graph have equal y-values, separating where one lies above the other.

📝 Essential Points

• In Example 7(c), the graphs intersect at x=0 and x=1 and the x-values are restricted by the domain −1≤x≤1.
• Squaring is justified in Example 7(c) because $x-1$ and $1-x^2$ are treated as non-negative quantities on the relevant domain.
• For Example 7(c), the solution to $|x-1|\ge \sqrt{1-x^2}$ is −1≤x≤0 or x=1.
• For Example 7(d), $|x-1|\ge\sqrt{1-x^2}$ holds only at the intersection point x=1 when the graphs are compared directly.

💡 Memory Hook

Absolute value (V) beats semicircle (hump) only where it sits above; check domain first, then use intersections to split the x-range.

📖 5. Circle and region inequalities

🔑 Key Concepts & Definitions

• Circle inequality : A circle inequality is a condition like $x^2+y^2 \le r^2$ or $x^2+y^2 \ge r^2$ that selects points relative to a circle of radius $r$.
• Inside circle region : The inside-the-circle region is the set of points satisfying $x^2+y^2 \le r^2$, including the boundary when the inequality is non-strict.
• Strict boundary inequality : A strict inequality like $x^2+y^2<r^2$ or $x<2$ excludes the boundary points where equality holds.
• Intersection of inequalities : The region satisfying several inequalities at once is the set of points that make every inequality true simultaneously.
• Region between curves : A region between curves is formed by combining two inequalities so the solution set lies above one curve and below another curve.

📝 Essential Points

• For a circle centred at the origin with radius 3, the inside-and-on condition is $x^2+y^2\le 9$, and combining with $x<2$ gives $x^2+y^2\le 9$ with points left of $x=2$.
• For the same circle, combining with $x+y\ge 3$ gives the region $x^2+y^2\le 9$ together with the half-plane on or above the line $x+y=3$.
• The intersection points for $y=x^2$ and $y=2x+3$ satisfy $x^2=2x+3$, giving $(−1,1)$ and $(3,9)$.
• The region defined by $y\ge x^2$ and $y\le 2x+3$ consists of points on or above the parabola and on or below the line, between $x=-1$ and $x=3$.
• For $y<2+x-x^2$, completing the square gives $y<2.25-(x+0.5)^2$, so it is below an upside-down parabola shifted right by $0.5$ and up by $2.25$.

💡 Memory Hook

Intersection needs two tests: “on/above lower curve” AND “on/below upper curve”; strict (<) means boundary is erased.

📖 6. Polynomial basics

🔑 Key Concepts & Definitions

• Degree of a polynomial : The degree of a polynomial is the highest power of x with a nonzero coefficient.
• Leading coefficient : The leading coefficient is the coefficient of the highest-power term in a polynomial.
• Monic polynomial : A monic polynomial is a polynomial whose leading coefficient equals 1.
• Real roots (real zeros) : Real roots, also called real zeros, are the real values of x that make P(x)=0.
• Polynomial equation degree n : A polynomial equation P(x)=0 has degree n when P(x) is a degree n polynomial.

📝 Essential Points

• A real polynomial P(x) is defined for all real x and is continuous and differentiable in x.
• A polynomial equation P(x)=0 of degree n has at most n real roots.
• If a polynomial includes a power of x that is not a positive integer, the expression is not a polynomial.
• Over a coefficient set, a polynomial is described by where its coefficients lie (e.g., integers, rationals, irrationals, or reals).
• When a polynomial is degree 4, it can have at most four real zeros.

💡 Memory Hook

Degree = highest x-power; monic ⇔ leading coefficient 1; real roots ⇔ P(x)=0.

📖 7. Polynomial operations and division

🔑 Key Concepts & Definitions

• Long division of polynomials : Long division of polynomials is an algorithm that produces a quotient and remainder when dividing one polynomial by another.
• Linear divisor restriction x≠−3 : A linear divisor restriction states the divisor cannot equal zero, so the division is only valid when x≠−3 for x+3.
• Dividend = Divisor × Quotient + Remainder : The dividend–divisor relationship states any polynomial division can be written as Dividend = Divisor × Quotient + Remainder.
• Remainder degree less than divisor degree : Remainder degree less than divisor degree means the remainder polynomial has smaller degree than the divisor polynomial.

📝 Essential Points

• For long division by (x+3), the quotient and remainder are found under the dividend using terms from the leading coefficients until subtraction leaves a final remainder.
• When the divisor is linear (first degree), the remainder is a constant because the remainder degree must be less than 1.
• When dividing P(x) by (x−3), you can check consistency by computing P(3) and using R=P(3) for the remainder.
• If the remainder is 0, then the quotient is exact and the division implies the divisor is a factor of the dividend.
• When writing long division, include missing dividend terms with zero coefficients (e.g., 4x^3+0x^2−19x+9).

💡 Memory Hook

Long division ends like a balance sheet: Dividend = Divisor×Quotient + Remainder, and the final R has lower degree than the divisor.

📖 8. Remainder and factor theorems

🔑 Key Concepts & Definitions

• Factor theorem : The factor theorem links polynomial zeros to factors: if P(a)=0 then (x−a) is a factor of P(x).
• Converse of factor theorem : The converse says that if (x−a) is a factor of P(x) then evaluating P at a gives P(a)=0.
• Remainder theorem : The remainder theorem relates division remainders to polynomial values, letting you use a remainder to test whether a would be a zero.
• Trial and error factoring : Trial and error factoring is trying likely values of x to find a zero, then using division to reduce the polynomial’s degree.

📝 Essential Points

• When a polynomial is divided by (x−a) and the remainder is 0, then (x−a) is a factor of the polynomial.
• For any polynomial P(x), P(a)=0 if and only if (x−a) is a factor of P(x).
• If P is degree n and you divide by a linear factor (x−a), the quotient has degree n−1.
• After finding one factor, factorising the quotient further gives factors of the original polynomial.
• For a monic cubic, possible zeros come from the factors of the constant term, then a zero check identifies the correct linear factor.

💡 Memory Hook

Factor test: P(a)=0 exactly when (x−a) can be pulled out as a factor.

📖 9. Roots and coefficients

🔑 Key Concepts & Definitions

• Sum of roots : The sum of the roots is the value obtained by adding all solution values of a polynomial equation that have been named as roots.
• Product of roots : The product of the roots is the value obtained by multiplying the named root values together for the polynomial equation.
• Quadratic coefficient relationships : Quadratic coefficient relationships link a quadratic’s coefficients to the sum of its two roots and the product of its two roots.
• Cubic coefficient relationships : Cubic coefficient relationships link a cubic’s coefficients to the sum of its three roots, the sum of pairwise products, and the product of all three roots.

📝 Essential Points

• For a quadratic $ax^2+bx+c=0$ with roots $\alpha,\beta$, the sum is $\alpha+\beta=-\frac{b}{a}$ and the product is $\alpha\beta=\frac{c}{a}$.
• For a cubic $ax^3+bx^2+cx+d=0$ with roots $\alpha,\beta,\gamma$, the sum is $\alpha+\beta+\gamma=-\frac{b}{a}$, the pairwise-sum is $\alpha\beta+\alpha\gamma+\beta\gamma=\frac{c}{a}$, and the product is $\alpha\beta\gamma=-\frac{d}{a}$.
• For quadratic roots $\alpha,\beta$, the identity $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$ lets you find squared-sum from sum and product.
• For cubic roots $\alpha,\beta,\gamma$, the identity $\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\alpha\gamma+\beta\gamma)$ links squared-sum to sum and pairwise products.

💡 Memory Hook

Quadratic: sum goes with $-\frac{b}{a}$ and product goes with $\frac{c}{a}$; cubic adds the pairwise products term $\frac{c}{a}$ and uses $-\frac{d}{a}$ for the triple product.

📖 10. Multiple roots of polynomials

🔑 Key Concepts & Definitions

• Zero of multiplicity : A zero c of a polynomial has multiplicity r>1 when the factor (x−c) appears r times in its factorisation.
• Derived polynomial multiplicity drop : If c is a multiplicity r zero of P(x), then c becomes a multiplicity (r−1) zero of P′(x), then (r−2) of P′′(x), and so on.
• Multiple-root test via P′ : A multiple root c of P(x) must also be a root of P′(x), because differentiating reduces the multiplicity by 1.

📝 Essential Points

• If $P(x)=(x-c)^rS(x)$ with r>0 and S(c)≠0, then $P'(x)=(x-c)^{r-1}Q(x)$ for some polynomial Q(x), so c has multiplicity r−1 in P′(x).
• For a polynomial of degree n, $P'(x)$ has degree n−1 and $P''(x)$ has degree n−2, so repeated differentiation can be used to locate multiplicities.
• A polynomial can have a root of multiplicity 2 (a double root) only if that same x-value makes P′(x)=0.
• Example method for a double root: differentiate, solve $P'(x)=0$, then check which candidates actually satisfy P(x)=0 to determine the repeated root(s).

💡 Memory Hook

Multiplicity r ⇒ derivatives kill one copy each time: $r,r−1,r−2,...$ across $P,P',P'',...$.

📖 11. Polynomial functions

🔑 Key Concepts & Definitions

• General polynomial function : A polynomial function of degree n has the form $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ with $a_n\ne 0$, and it is defined for every real $x$.
• General linear function : A polynomial function with highest power 1 is written as $f(x)=ax+b$ with $a\ne 0$ and models straight-line graphs.
• Odd function : An odd function satisfies $f(-x)=-f(x)$, so its graph has 180° rotational symmetry about the origin.
• Even function : An even function satisfies $f(-x)=f(x)$, so its graph is symmetric about the y-axis.

📝 Essential Points

• For any polynomial $f(x)=a_nx^n+\cdots+a_0$ with $a_n\ne 0$, the graph is continuous and differentiable for all real $x$.
• For $f(x)=x^n$ with $n\ne 1$, the x-axis is a tangent at the origin, so the curve just touches at $(0,0)$.
• If $n$ is odd in $f(x)=x^n$, the ends of the graph go in opposite directions as $x$ increases or decreases.
• If $n$ is even in $f(x)=x^n$, the ends of the graph go in the same direction as $x$ increases or decreases.
• For cubic $f(x)=ax^3+bx^2+cx+d$ with $a\ne 0$, the x-axis crossings correspond to real roots and the sign of $f(x)$ changes at each simple crossing.

💡 Memory Hook

Odd: f(−x)=−f(x) → flip sign; Even: f(−x)=f(x) → same sign; $x^n$ tangent at 0 for $n\ne 1$.

📖 12. Graphs of polynomial functions

🔑 Key Concepts & Definitions

• Sign diagram : A sign diagram records whether a polynomial is positive, zero, or negative on each interval split by its real zeros.
• Crossing the x-axis : Crossing the x-axis is when a graph passes through a real zero, changing sign across that x-value.
• Touching the x-axis : Touching the x-axis is when a graph meets the x-axis at a real zero but does not cross, so the sign does not change.
• Odd end behavior : For a polynomial of odd degree, the two ends of the graph go in opposite directions.
• Even end behavior : For a polynomial of even degree, the two ends of the graph go in the same direction.

📝 Essential Points

• For the cubic with positive $x^3$ coefficient, the graph increases as $x$ increases except possibly between its turning points.
• When a polynomial meets the x-axis, it may cut it (single zero), touch it (double zero), or cut it at an inflection point (triple zero).
• If the polynomial is written in factors, the sign on each interval comes from multiplying the factor signs and using that zeros split the sign changes.
• For $f(x)=2x^3-x^2-13x-6=(x+2)(2x+1)(x-3)$, the graph crosses the y-axis at $(0,-6)$.
• For $f(x)=(x+2)(2x+1)(x-3)$, $f(x)>0$ on $(-2,-\tfrac{1}{2})$ and for $x>3$, and $f(x)<0$ on $(-\tfrac{1}{2},3)$.
• Near $x=-2$ and $x=-1$ in $y=(x+2)(x-1)^2/(x+1)$, the graph cuts the x-axis at those zeros like straight-line behavior, while at $x=1$ it has a stationary turning point.

💡 Memory Hook

Single zero: cut; double zero: touch; triple zero: cut at inflection (C-T-Cinf).

📊 Synthesis Tables

Inequality sign changes when transforming

Operation	Sign/direction rule	When it matters
Add/subtract same number	Direction does not change	Always
Multiply by positive number	Direction does not change	Always
Multiply by negative number	Inequality direction reverses	Always
Take reciprocals	Reverses direction only if both sides have the same sign	Same sign; not if signs differ
Squaring	Direction depends on which of a and b is larger	Safe to preserve only when both sides are positive (or non-negative for square roots step)

⚠️ Common Pitfalls & Confusions

1. When solving x(x−4)>0, treating it like an equation and only using x(x−4)=0 gives wrong solutions because you must use “product positive” (both factors same sign).
2. For reciprocals, reversing the inequality direction even when the two sides have different signs causes an error; the source says reversal only when both sides have the same sign.
3. Squaring an inequality without sign conditions can change the inequality direction because squaring removes sign information; preserving direction needs both sides positive (and square roots need non-negative).
4. Including excluded rational-denominator values (like x=2 in 1/(x−2)) as solutions is a common mistake; they must be marked with open circles.
5. After multiplying a rational inequality by something like (x−2), you must consider the sign of the multiplier; the course uses methods that avoid case-splitting by squaring a known-positive factor.
6. With absolute value inequalities, forgetting to split according to the absolute value definition (x≥a vs x<a) or to check the correct “above/below” relation on the graph leads to wrong intervals.
7. For polynomial graphs, confusing “cutting” vs “touching” the x-axis (single vs double zero) gives the wrong sign pattern and wrong inequality answers.

✅ Exam Checklist

1. State and apply the inequality direction rules for adding/subtracting, multiplying by positive/negative, and reciprocals (same sign vs different signs).
2. Decide when squaring an inequality preserves direction (both sides positive as in the source) and when it may not; use square roots only when both sides are non-negative.
3. Solve quadratic inequalities by using factor sign analysis (“both factors positive or both negative” for product >0) and include endpoints correctly for ≤ or ≥.
4. Use the graphical method for quadratic inequalities by identifying the x-values where the parabola is above/below the x-axis and carefully include touch/cut points for ≤/≥.
5. Solve rational inequalities by identifying critical values (excluded denominator values) and then solving the corresponding boundary equation to place open/filled circles.
6. Apply the “multiply by (x−2)^2” strategy (or other known-positive squared factor) to avoid reversing inequality direction when the original multiplier’s sign is unknown.
7. Solve absolute value and square root inequalities by using domain restrictions (square root defined only where inside is non-negative) and comparing the absolute-value graph with the semicircle graph at intersection points.
8. Describe circle and region inequalities using x^2+y^2 ≤ r^2 / < r^2 and intersection of inequalities (on/between curves) and compute intersection points by solving the simultaneous equalities.
9. For polynomial basics, give the degree, leading coefficient, monic definition, real zeros definition, and the fact that a degree-n polynomial has at most n real roots.
10. Use long division correctly in polynomial division: include missing terms with zero coefficients and apply Dividend = Divisor × Quotient + Remainder with the remainder degree < divisor degree.
11. Apply the remainder theorem and factor theorem precisely: R = P(a) when dividing by (x−a), and P(a)=0 iff (x−a) is a factor; then use trial and error factoring/quotient factoring.
12. Use coefficient–root relationships for quadratics and cubics (sum/product and pairwise-sum) and the multiple-root test via derivatives to determine repeated roots.