Dividing Polynomials; Remainder and Factor Theorems

Dividing Polynomials; Remainder and Factor Theorems

Long Division of Polynomials

Polynomial long division is a method for dividing a polynomial (the dividend) by another polynomial (the divisor), resulting in a quotient and possibly a remainder. This process is similar to long division with numbers.

• Step 1: Arrange both the dividend and the divisor in descending powers of the variable.

• Step 2: Divide the first term of the dividend by the first term of the divisor. The result is the first term of the quotient.

• Step 3: Multiply every term in the divisor by the first term of the quotient. Write the resulting product beneath the dividend, aligning like terms.

• Step 4: Subtract the product from the dividend.

• Step 5: Bring down the next term from the original dividend and write it next to the remainder to form a new dividend.

• Step 6: Repeat the process with the new dividend until the degree of the remainder is less than the degree of the divisor.

The Division Algorithm:

For polynomials f(x) (dividend) and d(x) (divisor), there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that:

$f(x) = d(x)q(x) + r(x)$

where the degree of r(x) is less than the degree of d(x).

Example: Divide $3x^3 - 7x^2 + 11x - 3$ by $x - 3$ using long division.

Additional info: The process involves dividing $3x^3$ by $x$, multiplying, subtracting, and repeating until the remainder's degree is less than 1.

Synthetic Division

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

• Step 1: Arrange the polynomial in descending powers, using 0 for any missing terms.

• Step 2: Write the value c from the divisor x - c to the left. To the right, list the coefficients of the dividend.

• Step 3: Bring down the leading coefficient to the bottom row.

• Step 4: Multiply c by the value just written in the bottom row. Write the product in the next column of the second row.

• Step 5: Add the values in this column and write the sum in the bottom row.

• Step 6: Repeat the multiplication and addition process for all columns.

• Step 7: The numbers in the last row give the coefficients of the quotient, with the final value as the remainder. The degree of the quotient is one less than the original dividend.

Example: Use synthetic division to divide $3x^3 - 7x^2 + 11x - 3$ by $x + 2$.

Additional info: For $x + 2$, use $c = -2$. List coefficients: 3, -7, 11, -3. Apply the synthetic division steps to find the quotient and remainder.

The Remainder Theorem

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

• Key Point: To find the remainder of f(x) divided by x - c, simply evaluate f(c).

Example: Given $f(x) = 3x^3 + 4x^2 - 5x + 3$, use the Remainder Theorem to find $f(-4)$.

Additional info: Substitute $x = -4$ into the polynomial and compute the value to obtain the remainder.

The Factor Theorem

The Factor Theorem is a special case of the Remainder Theorem. It states that x - c is a factor of f(x) if and only if f(c) = 0.

• Key Point: If f(c) = 0, then x - c is a factor of f(x).

• Key Point: If x - c is a factor, then c is a zero (root) of f(x).

Example: Solve the equation $15x^3 + 14x^2 - 3x - 2 = 0$ given that $-1$ is a zero of $f(x) = 15x^3 + 14x^2 - 3x - 2$.

Additional info: Since $f(-1) = 0$, $x + 1$ is a factor. Use synthetic division to factor further and solve for other zeros.

Summary Table: Division Methods and Theorems

Method/Theorem	Purpose	Key Steps/Result
Long Division	Divide any polynomial by another polynomial	Arrange, divide, multiply, subtract, repeat until remainder degree is less than divisor
Synthetic Division	Divide by linear divisors of the form x - c	Use coefficients and c, perform multiplication and addition steps
Remainder Theorem	Find remainder when dividing by x - c	Remainder is f(c)
Factor Theorem	Test if x - c is a factor	If f(c) = 0, then x - c is a factor