Polynomials and Synthetic Division

The Polynomial Division Algorithm

Polynomial division mirrors numerical division, relating a dividend ($P(x)$) to its divisor ($d(x)$), quotient ($Q(x)$), and remainder ($R$).

$$P(x) = d(x)Q(x) + R$$

The Division Algorithm expresses a polynomial as the product of its divisor and quotient plus any remaining value.

$P(x)$=Dividend polynomial(unitless)

$d(x)$=Divisor polynomial(unitless)

$Q(x)$=Quotient polynomial(unitless)

$R$=Remainder constant (if divisor is linear)(unitless)

$R = 0$

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The divisor is a factor of the dividend.

Standard Form: Both polynomials must be written in descending order of degree before starting.

Degree Constraint: For the algorithm to finish, the degree of the remainder must be less than the degree of the divisor.

Quotient Degree: When dividing a degree-$n$ polynomial by a degree-$m$ divisor, the quotient has degree $n - m$.

Synthetic Division: The Shortcut

Synthetic division is a streamlined method for dividing by a linear divisor of the form $x - c$, focusing solely on coefficients.

$$x - c \implies \text{Use } c \text{ in the synthetic array}$$

The constant $c$ is the value that makes the divisor zero ($x - c = 0$).

$c$=Zero of the divisor(constant)

$x - c$=Linear binomial divisor(unitless)

$x + 3$

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$c = -3$ (Always flip the sign of the constant term).

Placeholder Zero: If a power of $x$ is missing in the dividend, a $0$ must be inserted in the synthetic array.

Leading Coefficient: The first coefficient of the dividend is always dropped straight down to begin the process.

Limitation: Synthetic division only works when the divisor is linear with leading coefficient $1$. For divisors like $2x - 1$, use long division or divide all coefficients by $2$ first.

The Remainder and Factor Theorems

The Remainder Theorem allows calculating $P(c)$ without division, while the Factor Theorem identifies roots where the remainder is zero.

$$P(c) = R$$

Evaluating a function at $c$ gives the same result as the remainder of $P(x) \div (x - c)$.

$P(c)$=Polynomial evaluated at $c$(value)

$R$=Remainder from division(value)

$P(c) = 0$

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$(x - c)$ is a factor and $c$ is a root.

Synthetic Substitution: Using synthetic division to find $P(c)$ is often faster than direct substitution for high-degree polynomials.

Zero of a Function: If $(x - c)$ is a factor, then $x = c$ is an x-intercept on the graph of $P(x)$.

Converse Direction: The Factor Theorem works both ways — $P(c) = 0$ proves $(x - c)$ is a factor, and $(x - c)$ being a factor proves $P(c) = 0$.

Finding Unknown Coefficients: Set $P(c) = R$ (given remainder) to build equations for unknown parameters like $k$ or $p$.

Depressed Equations and Complete Factoring

When a known root $c$ is used in synthetic division, the quotient polynomial (one degree lower) is called the depressed equation. Solving it finds the remaining roots.

$$P(x) = (x - c) \cdot Q(x) \implies \text{solve } Q(x) = 0$$

After removing one factor via synthetic division, the depressed equation $Q(x) = 0$ contains the remaining roots.

$Q(x)$=Depressed polynomial (quotient after division)(unitless)

$(x - c)$=Factor corresponding to known root $c$(unitless)

Multiple known roots

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Apply synthetic division repeatedly — each division reduces the degree by 1.

Depressed equation is quadratic

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Factor directly or use the quadratic formula to find remaining roots.

Degree Reduction: A degree-$n$ polynomial with $k$ known roots requires synthetic division $k$ times, leaving a degree-$(n - k)$ depressed equation.

Order Independence: Multiple synthetic divisions can be performed in any order — the final depressed equation is the same.

Forming Polynomial Equations from Roots

Given all roots of a polynomial, the equation can be reconstructed by multiplying the corresponding linear factors together.

$$P(x) = (x - r_1)(x - r_2) \cdots (x - r_n)$$

Each root $r_i$ contributes a factor $(x - r_i)$. Their product gives the complete polynomial.

$r_1, r_2, \ldots, r_n$=The $n$ roots of the polynomial(unitless)

$(x - r_i)$=Linear factor for root $r_i$(unitless)

Quadratic ($n = 2$)

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$x^2 - (r_1 + r_2)x + r_1 r_2 = 0$, i.e., $x^2 - Sx + P = 0$ where $S$ = sum and $P$ = product of roots.

Sum and Product Shortcut: For a quadratic with roots $\alpha, \beta$: the equation is $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.

Scaling: Any constant multiple $k \cdot P(x) = 0$ has the same roots. The leading coefficient can be chosen freely.