Additional Mathematics Synthesis Guide 24: Cubic Polynomials, Factor and Remainder Theorems, Roots, Multiplicity and Graph Structure

Polynomial → remainder test → factor → quotient → roots → multiplicity → graph behaviour → verification.

1. The factor theorem

For a polynomial P(x),

x−a is a factor of P(x) if and only if P(a)=0.

This turns a divisibility statement into a function-value test. If P(3)=0, then the graph y=P(x) passes through the x-axis at x=3 and the algebra contains a factor x−3.

2. The remainder theorem

When P(x) is divided by x−a, the remainder is P(a).

This is broader than the factor theorem. A remainder of zero gives a factor; a non-zero remainder gives exact information about the division without carrying out the whole division.

For example, if P(x)=x³−2x²+5x−7, the remainder on division by x−2 is

P(2)=8−8+10−7=3.

3. Worked factorisation from a known root

Factorise

P(x)=x³−6x²+11x−6.

Test x=1:

P(1)=1−6+11−6=0.

So x−1 is a factor. Divide P(x) by x−1 to obtain

x²−5x+6.

Then

x²−5x+6=(x−2)(x−3).

Hence

P(x)=(x−1)(x−2)(x−3).

4. Root-factor correspondence

From the factorisation above, the roots are 1,2,3. The graph crosses the x-axis at each root because each factor appears to the first power.

The statement “x=a is a root” and “x−a is a factor” are two descriptions of the same algebraic structure.

5. Polynomial division as a structural reduction

Once a linear factor is known, cubic division reduces the problem to a quadratic. This is important because quadratic structure is already well understood: factorisation, discriminant, completing the square and root formulas become available.

A cubic question often becomes manageable not by solving a general cubic formula, but by locating one factor and transferring the remaining work to a quadratic.

6. Parameter from a factor condition

Let

P(x)=x³+kx²−5x−6.

If x−2 is a factor, then P(2)=0:

8+4k−10−6=0.

Thus 4k−8=0, so

k=2.

The factor condition turned an unknown coefficient into a linear equation.

7. Two remainder conditions can determine two parameters

Let

P(x)=x³+ax+b.

Suppose the remainder on division by x−1 is 4 and the remainder on division by x+1 is 0.

Then

P(1)=1+a+b=4,

P(−1)=−1−a+b=0.

So a+b=3 and −a+b=1. Hence

b=2, a=1.

8. Repeated roots and multiplicity

If

P(x)=(x−a)²(x−b),

then x=a is a root of multiplicity 2 and x=b has multiplicity 1.

A repeated factor changes graph behaviour. At a simple root the graph normally crosses the axis. At a double root the graph touches the axis and turns back, because the sign of the squared factor does not change through the root.

9. Worked double-root example

Consider

P(x)=(x−2)²(x+1).

The roots are x=2 (double) and x=−1 (simple).

Near x=2, (x−2)² remains non-negative while x+1 remains positive, so P(x) is non-negative on both sides of x=2. The graph touches the x-axis there rather than crossing.

At x=−1, the factor x+1 changes sign and the other factor is positive, so the graph crosses.

10. A repeated root also satisfies the derivative equation

If P(x)=(x−a)²Q(x), then differentiating shows P′(a)=0. Therefore a repeated root satisfies

• P(a)=0
• P′(a)=0

This provides a powerful parameter method when a problem states that a polynomial has a repeated root.

11. Parameter problem with a repeated root

Let

P(x)=x³−3x²+kx−4.

Suppose x=2 is a repeated root. Then P(2)=0 gives

8−12+2k−4=0,

so 2k−8=0 and k=4.

Check the derivative:

P′(x)=3x²−6x+k.

At x=2 and k=4, P′(2)=12−12+4=4, not zero. Therefore the original statement cannot be satisfied by any k: the factor condition alone gave a candidate, but the repeated-root condition rejects it.

This is a useful example of why multiplicity needs more information than ordinary factorhood.

12. Build a cubic with a specified repeated root

If a monic cubic has a double root 2 and another root −3, then

P(x)=(x−2)²(x+3).

Expand:

P(x)=(x²−4x+4)(x+3)=x³−x²−8x+12.

Coefficient information can therefore be reconstructed directly from root multiplicities.

13. Sum and product of roots as verification

For a monic cubic

x³+Ax²+Bx+C

with roots r₁,r₂,r₃:

• r₁+r₂+r₃=−A
• r₁r₂+r₁r₃+r₂r₃=B
• r₁r₂r₃=−C

For x³−6x²+11x−6, the roots 1,2,3 have sum 6, pairwise sum 11 and product 6, matching the coefficients.

These relationships are excellent checks after factorisation.

14. Rational-root search can be bounded

For a monic polynomial with integer coefficients, any integer root must divide the constant term. Thus for

x³−4x²−7x+10,

possible integer roots are ±1,±2,±5,±10.

Testing a short justified list is better than guessing indefinitely. Here P(1)=0, so x−1 is a factor.

15. Continue factorisation after one root

Divide

x³−4x²−7x+10

by x−1 to obtain

x²−3x−10.

This factors as (x−5)(x+2). Therefore

P(x)=(x−1)(x−5)(x+2).

The roots are −2,1,5.

16. End behaviour comes from the leading term

For a cubic ax³+lower-order terms:

• if a>0, y→∞ as x→∞ and y→−∞ as x→−∞;
• if a<0, the directions reverse.

The lower-order terms change the detailed shape and intercepts but not the far-end direction.

17. Cubics must have at least one real root

A real cubic has opposite end behaviour at x→∞ and x→−∞. Because a polynomial is continuous, the graph must cross the x-axis at least once.

It may have one real root and two non-real roots, or three real roots counting multiplicity.

18. Derivative reveals turning points

For

P(x)=x³−3x²−9x+5,

the derivative is

P′(x)=3x²−6x−9=3(x−3)(x+1).

Stationary points occur at x=−1 and x=3. A cubic can therefore have up to two stationary points because its derivative is quadratic.

If the derivative quadratic has no real roots, the cubic is monotonic and has only one real x-axis crossing.

19. Multiplicity three gives a flattened crossing

For

P(x)=(x−2)³,

the root x=2 has multiplicity 3. The graph crosses the axis because an odd-power factor changes sign, but it does so with horizontal tangent because

P′(x)=3(x−2)²

and P′(2)=0.

This differs from a simple crossing and from a double-root touch.

20. Multiplicity predicts axis behaviour

• Odd multiplicity: the polynomial changes sign through the root, so the graph crosses the axis.
• Even multiplicity: the sign does not change, so the graph touches and turns.
• Multiplicity greater than 1: the derivative is also zero at the root, creating flattened contact.

These statements come from the local factor (x−a)^m multiplied by a non-zero nearby factor.

21. Reverse problem from graph information

A monic cubic crosses the x-axis at x=−2 and touches it at x=1. Then the roots and multiplicities imply

P(x)=(x+2)(x−1)².

Expand:

P(x)=x³−3x+2.

The missing x² coefficient is meaningful: the roots −2,1,1 sum to zero.

22. Remainder data can test a factorisation

Suppose a proposed factorisation of P(x) is available. Evaluate both forms at a convenient value not equal to a root. If the results disagree, the factorisation is wrong.

For example, if P(x)=(x−1)(x−2)(x−3), then P(0)=−6. The expanded form x³−6x²+11x−6 also gives −6.

This does not prove equality by itself, but it is a cheap error detector. Full expansion or coefficient comparison provides exact verification.

23. A cubic equation may hide a quadratic after one factor

Once one root a is found, write

P(x)=(x−a)Q(x)

with Q quadratic. Then the number and nature of the remaining real roots are controlled by the discriminant of Q.

This creates a direct bridge from cubic structure back to the quadratic methods of Guide 21.

24. Common failure patterns

• Using P(a)=0 as though it proves a repeated root.
• Finding one factor and forgetting to factorise or solve the quadratic quotient.
• Testing random integer roots without using the constant-term divisibility constraint.
• Ignoring multiplicity when interpreting graph crossings.
• Assuming every stationary point of a cubic lies on the x-axis.
• Using a graph sketch to invent roots not supported by exact algebra.
• Losing a sign during polynomial division and not checking by recombination.
• Forgetting coefficient/root relationships as a verification tool.

25. A reliable cubic routine

1. Use factor/remainder information before expanding or solving.
2. When integer coefficients are present, bound possible integer roots using divisors of the constant term.
3. Once a root is found, divide to reduce the cubic to a quadratic.
4. Factor or solve the quadratic quotient completely.
5. Track multiplicities of repeated factors.
6. Use P(a)=P′(a)=0 when repeated-root information is given.
7. Check root sums/products against coefficients.
8. Use derivative and multiplicity to support the graph sketch, not replace exact root work.

26. Practice set

1. Find the remainder when x³−2x²+4x−5 is divided by x−2.
2. Determine whether x−1 is a factor of x³−4x²+x+2.
3. Factorise x³−6x²+11x−6.
4. Factorise x³−4x²−7x+10.
5. Find k if x−2 is a factor of x³+kx²−5x−6.
6. For P(x)=x³+ax+b, P(1)=4 and P(−1)=0. Find a,b.
7. State the root multiplicities of (x−3)²(x+2).
8. Describe how its graph meets the x-axis at each root.
9. Expand (x−2)²(x+3).
10. For x³−x²−8x+12, verify the roots 2,2,−3 using coefficient sums and products.
11. A monic cubic has roots 1,1,5. Find the polynomial.
12. A monic cubic crosses at x=−1 and touches at x=4. Find the polynomial.
13. Why must a repeated root a satisfy P′(a)=0?
14. Find the stationary x-values of x³−3x²−9x+5.
15. How many stationary points can a real cubic have at most?
16. Describe the x-axis behaviour of (x−1)³.
17. Why does a real cubic have at least one real root?
18. Give the possible integer roots of x³−5x²+2x+8.
19. If one root of a cubic is known, what mathematical object controls the remaining two roots after division?
20. Give one independent verification after factorising a cubic completely.

Answers

1. P(2)=8−8+8−5=3.
2. P(1)=1−4+1+2=0, so yes.
3. (x−1)(x−2)(x−3).
4. (x−1)(x−5)(x+2).
5. k=2.
6. a=1,b=2.
7. x=3 has multiplicity 2; x=−2 has multiplicity 1.
8. Touches and turns at x=3; crosses at x=−2.
9. x³−x²−8x+12.
10. Sum 2+2−3=1, matching −(−1)=1; pairwise sum 4−6−6=−8; product −12, matching −12.
11. (x−1)²(x−5)=x³−7x²+11x−5.
12. (x+1)(x−4)².
13. Because P contains (x−a)², so differentiation leaves a factor x−a in every term at x=a.
14. P′=3(x−3)(x+1), so x=−1,3.
15. Two.
16. Crosses with a horizontal tangent/flattened crossing at x=1.
17. Its continuous graph has opposite signs at sufficiently large positive and negative x because the leading cubic term dominates.
18. ±1,±2,±4,±8.
19. A quadratic quotient, whose discriminant/factorisation determines the remaining roots.
20. Expand the factors and compare coefficients, or check root sum/product relationships.

27. What mastery looks like

Mastery means the learner uses remainder and factor information efficiently, reduces a cubic to a quadratic after finding one root, distinguishes simple and repeated roots, connects multiplicity to graph behaviour, uses derivative conditions for repeated roots, and verifies complete factorisations against coefficients.

The transfer test is to supply mixed evidence — a remainder, a known root, a repeated contact on a graph, and one unknown coefficient. If the learner can combine these without treating them as separate chapter tricks, cubic-polynomial reasoning has become structural.

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Continue through Batch 06

• Guide 21: Quadratic Functions, Completing the Square, Global Bounds and Modelling
• Guide 22: Simultaneous Equations, Line–Curve Intersections, Discriminants and Tangency Conditions
• Guide 23: Binomial Theorem, General Terms, Coefficient Extraction and Parameter Problems
• Additional Mathematics Directory
• BTT Mathematics Hub