Polynomial Division, Remainder Theorem, and Synthetic Division

Polynomial Division and Factors

Dividing Polynomials and Identifying Factors

When dividing a polynomial P(x) by another polynomial d(x), if the remainder is zero, then d(x) is a factor of P(x). This concept is fundamental in algebra and precalculus for factoring and solving polynomial equations.

• Factor: A polynomial d(x) is a factor of P(x) if P(x) divided by d(x) leaves a remainder of zero.

• Example: Determine if d(x) = x + 1 or d(x) = x - 3 are factors of P(x) = x^3 + 2x^2 - 5x - 6 by performing polynomial division.

Polynomial Division Algorithm

Division Algorithm for Polynomials

When dividing a polynomial P(x) by a nonzero polynomial d(x), the result can be expressed as:

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

• Q(x): The quotient polynomial.

• R(x): The remainder polynomial, where the degree of R(x) is less than the degree of d(x).

• Formula: $\frac{P(x)}{d(x)} = Q(x) + \frac{R(x)}{d(x)}$

• Note: Q(x) must have degree less than P(x), and R(x) must have degree less than d(x).

Remainder Theorem

Theorem and Application

The Remainder Theorem states that if a polynomial f(x) is divided by x - c, the remainder is f(c).

• Theorem: If r is the remainder obtained from dividing f(x) by x - c, then f(c) = r.

• Proof Outline: For some polynomial Q(x), $f(x) = (x - c)Q(x) + r$. Substituting x = c gives $f(c) = r$.

• Example: To find the remainder when f(x) is divided by x - 3, compute f(3).

Synthetic Division

Efficient Division Method

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x - c. It is faster and more streamlined than long division for this specific case.

• Procedure: Write the coefficients of P(x) and use c from x - c to perform the synthetic division steps.

• Result: The last number in the synthetic division process is the remainder; the other numbers give the coefficients of the quotient polynomial.

• Example: Divide 2x^3 + 7x^2 - 5 by x + 3 using synthetic division.

Evaluating Polynomials and Finding Solutions

Evaluating at Specific Values

To evaluate a polynomial g(x) at a specific value, substitute the value for x and compute the result.

• Example: Find g(-5), g(3), g(i + 1), g(-i) for a given polynomial g(x).

Finding Other Solutions

If a value is a solution to P(x) = 0, use polynomial division to find other solutions.

• Example: If P(x) = x^4 - 7x^2 + 16x - 12 has a known solution, divide by the corresponding factor to find the remaining solutions.

Summary Table: Polynomial Division Concepts

Concept	Description	Key Formula
Factor	If P(x) divided by d(x) leaves remainder 0, d(x) is a factor.	$P(x) = d(x)Q(x)$
Division Algorithm	Expresses division as quotient and remainder.	$P(x) = d(x)Q(x) + R(x)$
Remainder Theorem	Remainder when dividing by x - c is f(c).	$f(c) = r$
Synthetic Division	Shortcut for dividing by linear factors.	Coefficients manipulated using c

Additional info: The notes cover essential Precalculus topics on polynomial division, factorization, the remainder theorem, and synthetic division, with examples and definitions suitable for exam preparation.