5.1 Remainder and Factor Theorem

Polynomial Division

When a polynomial $P(x)$ of degree $n$ is divided by a divisor $D(x)$ of degree less than or equal to $n$, we obtain:

$P(x)=D(x)\cdot q(x)+r(x)$

where $q(x)$ is the quotient and $r(x)$ is the remainder, with $\mathrm{deg}(r)<\mathrm{deg}(D)$.

Division by a Linear Polynomial

When dividing by a linear polynomial $(x-a)$, the remainder $r$ is a constant (degree 0):

$P(x)=(x-a)q(x)+r$

Example: Divide $P(x)=2x^3-3x^2+x-5$ by $(x-2)$.

Using synthetic division with $a=2$:

| 2 | 2 | −3 | 1 | −5 | |---|---|----|----| | | | 4 | 2 | 6 | | | 2 | 1 | 3 | 1 |

So $q(x)=2x^2+x+3$ and $r=1$.

Verification: $P(2)=2(8)-3(4)+2-5=16-12+2-5=1$ ✓

Division by a Quadratic Polynomial

When dividing by a quadratic $D(x)=x^2+bx+c$, the remainder is at most linear: $r(x)=Ax+B$.

Example: Divide $P(x)=x^4-2x^3+3x-1$ by $x^2-x+1$ using long division.

$x^4-2x^3+3x-1=(x^2-x+1)(x^2-x-1)+(3x-2)$

Quotient: $x^2-x-1$, Remainder: $3x-2$.

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Remainder Theorem

Proof: From the division algorithm: $P(x)=(x-a)q(x)+r$ Substituting $x=a$: $P(a)=(a-a)q(a)+r=0+r=r$ Therefore $r=P(a)$. $■$

Key Point: For divisor $(x+a)=(x-(-a))$, substitute $x=-a$.

Worked Example

Find the remainder when $P(x)=3x^3-4x^2+2x-1$ is divided by $(x+2)$.

Solution: Here $a=-2$. $P(-2)=3(-8)-4(4)+2(-2)-1=-24-16-4-1=-45$

Remainder $=-45$.

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Factor Theorem

The Factor Theorem is a special case of the Remainder Theorem: if the remainder $P(a)=0$, then $(x-a)$ divides $P(x)$ exactly.

Converse: If $(x-a)$ is a factor of $P(x)$, then $P(a)=0$.

Factorizing a Cubic Polynomial

Steps:

1. Find a value $a$ such that $P(a)=0$ (try $\pm 1,\pm 2,\pm 3,\dots$, factors of the constant term).
2. Divide $P(x)$ by $(x-a)$ to get a quadratic quotient $q(x)$.
3. Factorize $q(x)$ by inspection or the quadratic formula.

Example: Factorize $P(x)=x^3-6x^2+11x-6$.

Step 1: Test $x=1$: $P(1)=1-6+11-6=0$ ✓ So $(x-1)$ is a factor.

Step 2: Divide by $(x-1)$: $x^3-6x^2+11x-6=(x-1)(x^2-5x+6)$

Step 3: Factorize $x^2-5x+6=(x-2)(x-3)$.

$\therefore P(x)=(x-1)(x-2)(x-3)$

Zeros: $x=1,2,3$.

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Real-World Applications

1. Geometry — Volume and Dimensions

If the volume of a rectangular box is given by a polynomial $V(x)$ and one dimension (height) is a linear factor $H(x)=(x-a)$, then the floor area is:

$\text{Area}=Q(x)=\frac{V(x)}{H(x)}$

This requires $H(x)$ to be a factor of $V(x)$, verified using the Factor Theorem: $V(a)=0$.

Example: $V(x)=x^3-6x^2+11x-6$, height $=(x-1)$.

Since $V(1)=0$, divide: $\text{Area}=x^2-5x+6=(x-2)(x-3)$.

2. Polynomial Regression

In data science, polynomial regression fits a polynomial $P(x)$ to data. The Remainder Theorem helps evaluate $P(x)$ at specific data points efficiently.

3. Signal Processing

Polynomials model signals. Factorization using the Factor Theorem identifies frequencies (zeros) where the signal vanishes.

4. Coding Theory

Error-detecting codes use polynomial division. The remainder when a message polynomial is divided by a generator polynomial is appended as a checksum.

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Summary Table