3.2 Dividing Polynomials

3.2.1 Learning Objectives

By the end of this section, you should be able to:

divide polynomials using long division;
divide by a linear expression using synthetic division;
express a division result using the Division Algorithm;
evaluate a polynomial using the Remainder Theorem; and
use the Factor Theorem to factor polynomials and construct polynomial functions.

3.2.2 Polynomial Long Division

Polynomial long division follows the same structure as numerical long division. Arrange both polynomials in descending powers, inserting zero-coefficient terms for any missing powers.

At each stage:

divide the leading term of the current dividend by the leading term of the divisor;
multiply the divisor by that quotient term;
subtract;
bring down the next term; and
repeat until the remainder has lower degree than the divisor.

Example 3.6 Dividing by a Binomial

Divide \(2x^3+3x^2-11x-6\) by \(x+3\).

Solution

The first quotient term is \(2x^2\). Continue multiplying and subtracting:

\[ \begin{array}{r|rrrr} & 2x^2 & -3x & -2\\ x+3 & 2x^3 & +3x^2 & -11x & -6\\ - & \underline{(2x^3} & \underline{+6x^2)}\\ & & -3x^2 & -11x\\ & -& \underline{ (-3x^2 } & \underline{ -9x) }\\ & & & -2x & -6\\ & & -& \underline{ ( -2x } & \underline{ -6 )}\\ & & & & 0 \end{array} \]

\[ \boxed{\frac{2x^3+3x^2-11x-6}{x+3}=2x^2-3x-2} \]

Exercise 3.6 Divide \(x^3+4x^2+x-6\) by \(x+2\).

3.2.3 The Division Algorithm

For polynomials \(P(x)\) and \(D(x)\), with \(D(x)\ne0\), there are unique polynomials \(Q(x)\) and \(R(x)\) such that

\[ P(x)=D(x)Q(x)+R(x), \]

where \(R(x)=0\) or the degree of \(R\) is less than the degree of \(D\). These are the dividend, divisor, quotient, and remainder.

Example 3.7 Writing a Division Statement

Divide \(x^3-2x^2+4x+5\) by \(x^2+1\) and write the result using the Division Algorithm.

Solution

Long division gives quotient \(x-2\) and remainder \(3x+7\). Therefore,

\[ \boxed{x^3-2x^2+4x+5=(x^2+1)(x-2)+(3x+7)}. \]

Exercise 3.7 If dividing \(P(x)\) by \(x-4\) gives quotient \(x^2+3\) and remainder \(5\), write \(P(x)\) in expanded form.

3.2.4 Synthetic Division

Synthetic division is a shortened method for division by a linear divisor of the form \(x-c\). Write only the coefficients of the dividend, including zeros for missing powers. Use \(c\), not the constant appearing after the minus sign.

Example 3.8 Using Synthetic Division

Divide \(2x^4-3x^3+0x^2+5x-7\) by \(x-2\).

Solution

\[ \begin{array}{r|rrrrr} 2 & 2 & -3 & 0 & 5 & -7\\ & & 4 & 2 & 4 & 18\\ \hline & 2 & 1 & 2 & 9 & 11 \end{array} \]

The quotient coefficients are \(2,1,2,9\), and the remainder is \(11\).

\[ \boxed{2x^3+x^2+2x+9+\frac{11}{x-2}} \]

Exercise 3.8 Use synthetic division to divide \(x^3-5x+6\) by \(x+2\).

3.2.5 The Remainder and Factor Theorems

The Remainder Theorem states that the remainder when \(P(x)\) is divided by \(x-c\) is \(P(c)\). This provides an efficient way to evaluate a polynomial.

The Factor Theorem follows immediately:

\[ x-c\text{ is a factor of }P(x)\quad\Longleftrightarrow\quad P(c)=0. \]

Example 3.9 Testing a Possible Factor

Determine whether \(x-3\) is a factor of \(P(x)=2x^3-5x^2-4x+3\).

Solution

\[ \begin{aligned} P(3)&=2(3)^3-5(3)^2-4(3)+3\\ &=54-45-12+3\\ &=0. \end{aligned} \]

Because the remainder is zero,

\[ \boxed{x-3\text{ is a factor of }P(x)}. \]

Exercise 3.9 Find the remainder when \(P(x)=x^4-2x+5\) is divided by \(x+1\).

3.2.6 Factoring and Constructing Polynomials

Once a known zero is found, synthetic division reduces the degree and may reveal additional factors. Conversely, specified zeros determine linear factors. A leading coefficient can then be included to construct a particular polynomial.

Example 3.10 Factoring with a Known Zero

Given that \(2\) is a zero of \(P(x)=x^3-3x^2-4x+12\), factor \(P(x)\) completely.

Solution

Synthetic division by \(x-2\) gives the quotient \(x^2-x-6\):

\[ P(x)=(x-2)(x^2-x-6). \]

Factor the quadratic:

\[ x^2-x-6=(x-3)(x+2). \]

\[ \boxed{P(x)=(x-2)(x-3)(x+2)} \]

Exercise 3.10 Write a monic cubic polynomial with zeros \(-2\), \(1\), and \(4\).

3.2.7 Conceptual Takeaways

Polynomial division reverses polynomial multiplication.
The remainder must have lower degree than the divisor.
Synthetic division is efficient but applies only to divisors \(x-c\).
Evaluating \(P(c)\) gives the remainder from division by \(x-c\).
A zero, an \(x\)-intercept, and a linear factor are three forms of the same relationship.

3.2.8 Skills You Should Be Able to Do

Divide polynomials by long division.
Use synthetic division with correct coefficients and signs.
Write a result in Division Algorithm form.
Apply the Remainder and Factor Theorems.
Factor or construct a polynomial from known zeros.

3.2.9 Practice Problems with Solutions

Divide \(x^3+2x^2-5x-6\) by \(x+3\).

Show Solution

Synthetic division uses \(-3\):

\[ \begin{array}{r|rrrr} -3 & 1 & 2 & -5 & -6\\ & & -3 & 3 & 6\\ \hline & 1 & -1 & -2 & 0 \end{array} \]

Therefore, \(\boxed{x^2-x-2}\).
Divide \(2x^3+x^2-7x+4\) by \(x-2\).

Show Solution

Using synthetic division with \(2\) gives coefficients \(2,5,3\) and remainder \(10\). Thus,

\[ \boxed{2x^2+5x+3+\frac{10}{x-2}}. \]
Divide \(x^4-5x^2+3x-2\) by \(x^2-1\).

Show Solution

Include the missing cubic term. Long division gives quotient \(x^2-4\):

\[ (x^2-1)(x^2-4)=x^4-5x^2+4. \]

Subtracting from the dividend leaves \(3x-6\). Therefore,

\[ \boxed{x^2-4+\frac{3x-6}{x^2-1}}. \]
Find the remainder when \(P(x)=3x^4-2x^2+x-8\) is divided by \(x-2\).

Show Solution

By the Remainder Theorem,

\[ P(2)=3(16)-2(4)+2-8=34. \]

The remainder is \(\boxed{34}\).
Determine whether \(x+2\) is a factor of \(P(x)=x^3+3x^2-4\).

Show Solution

Evaluate at \(-2\):

\[ P(-2)=-8+12-4=0. \]

Therefore, \(\boxed{x+2\text{ is a factor}}\).
Factor \(P(x)=x^3+x^2-10x+8\), given that \(x=1\) is a zero.

Show Solution

Synthetic division by \(x-1\) gives \(x^2+2x-8\). Then

\[ x^2+2x-8=(x+4)(x-2). \]

Hence,

\[ \boxed{P(x)=(x-1)(x+4)(x-2)}. \]
Write a degree-\(4\) polynomial with leading coefficient \(-2\), zeros \(0\), \(3\), and \(-1\), where \(-1\) has multiplicity \(2\).

Show Solution

The factors are \(x\), \(x-3\), and \((x+1)^2\). Including the leading coefficient gives

\[ \boxed{P(x)=-2x(x-3)(x+1)^2}. \]
When \(P(x)\) is divided by \(x-2\), the quotient is \(x^2+3x-1\) and the remainder is \(5\). Find \(P(x)\).

Show Solution

Use the Division Algorithm:

\[ \begin{aligned} P(x)&=(x-2)(x^2+3x-1)+5\\ &=x^3+x^2-7x+2+5\\ &=x^3+x^2-7x+7. \end{aligned} \]

\[ \boxed{P(x)=x^3+x^2-7x+7} \]