Factoring and Operations on Polynomials

Polynomials

A polynomial is usually described by the number of terms it contains and its degrees.

Different polynomials are described based on the number of terms.

Coefficient is a constant by which a variable is multiplied.

Leading coefficient is the coefficient of the leading (highest) term in a given polynomial.

The degree of the polynomial is equal to the highest degree (of any term) in the polynomial.
The first three are special polynomials.

Example 1: The degree of 3t is 1
Example 2: 4x⁴+3x²+2x
– Degree of the given polynomial is 4
– The coefficient of: x⁴ is 4, x² is 2, and x is 2
– The leading coefficient is 4
Note: The degree of constant is zero

Addition of Polynomials

1. Combine all the like terms.
2. Add all the like terms together.

Note: Like terms must contain the same variable(s) raised to the same power(s).

Example: Add

Solution:

Try this: Add (5ab+3ab²+7a) and (8b+6a²b+2ab+3a)

Subtraction of Polynomials

1. Place parentheses around the expression being subtracted.
2. Remove the parentheses by distributing the minus sign. This will change the sign of each term
of the polynomial being subtracted.
3. Add the polynomial by combining like terms.

Example: Subtract (3x²+2x+3) from (5x²+6x+2)

Solution: Step1

Try this: Subtract (-2a² – 6a +10) from (a²+8)

Multiplication of Polynomials

Multiplying a Monomial by a Monomial

1. Multiply the numerical coefficients to find the coefficient of the product.
2. Multiply the variables, simplifying powers with the same base.

Example: Multiply (4x) by (3x²)
Solution: 4x * 3x² = (4*3)(x*x²) = 12x³

Try This: Multiply (3 a³b²) by (-2ab)

Multiplying a Monomial by a Polynomial

Distribute the monomial over the polynomial by multiplying the monomial by every term in the
polynomial and joining the products by their proper sign.

Example: Multiply (3x – 4) by 5
Solution: 5(3x – 4) = 15x – 20

Multiplying a Polynomial by a Polynomial

Multiply each term of the first polynomial by each term of the second polynomial, then combine
like terms.
Example: Multiply (5x-7) by (6x + 4)
Solution: 5x * 6x + 5x * 4 – 7 * 6x – 7 * 4 = 30x² + 20x – 42x – 28 = 30x² – 22x +28.

Try this: Multiply (3x² + 2x) by (4x+ 3)

Factoring

A product is found by multiplying numbers or terms together. The numbers or terms multiplied
together are called the factors of the product. A factor may consist of more than one term.

Example: 5(x+y) = 5x + 5y
One factor is 5 and second is (x+y)

Factoring Polynomials with a Common Monomial Factor

1. Factor out the greatest common factor(GCF) from each term of the polynomial.
2. Write the two factors as the indicated product.

Example: 3a⁴ + 6a³ + 3a²
Solution: GCF is 3a²
3a²(a²+3a+1)

Try This: Factor (a²b² – ab)

Factoring Equations of the Form x²+bx+c

Example: Factor x² + 7x + 12

Solution: The first term in each binomial factor is x.
x² + 7x + 12
(x ) (x )
Search for two factors of 12 that have a sum of 7
Factors of 12
(1)(12)
(2)(6)
(3)(4) →These are the factors of 12 that have the sum as 7.
Choose 3 and 4 since 3*4 = 12 and 3+4 = 7
Factor by using the correct combination of terms
x² + 7x + 12 = (x +3) (x+4)
Final answer is (x+3) (x+4)
Check: (x+3) (x+4) = x² + 7x + 12

Try this: Factor x² -5x + 6

Factoring Equations of the Form ax² + bx +c

Example: Factor 5x² + 21x + 4
Solution: Write all possible factors of 5 and 4
5x² + 21x + 4

Search for the combination that will yield a sum of 21.

Since the correct combinations are 1 and 5, and 4 and 1, we can now
factor accordingly.

Final answer is (x + 4) (5x + 1)

Try This: Factor: 4x² – 12x + 9

Division of Polynomials

Divisor could be defined as a quantity or variable that divides another. Dividend is simply the
number or quantity that is to be divided.

For example, when we divide (5x-7) by (6x + 4), (6x+4) is the divisor and (5x-7) is the dividend.

Long Division
1. Write the problem in long division form.
a. Arrange the divisor and dividend in descending order .
b. Account for missing terms.
2. To find the terms of the quotient:
a. Divide the first term in the divisor into the first term in the dividend; write the answer as the first term in the quotient.
b. Multiply the divisor by the first term in the quotient, writing the product under the dividend.
c. Subtract like terms.
d. Bring down the next term.
3. Repeat (2) divide, multiply, subtract and bring down.
4. Continue until the reminder has a lower degree than divisor.

Example: Divide (x² + 5x + 6) by (x + 2)
Solution: 1. Write in long division form

2. Find the first term in the quotient,

a. What must you multiply ‘x’ by to get ‘x²’?

b. Multiply (x) times (x+2)

c. Subtract like terms

d. Bring down the next term

3. Find the second term in the quotient,

a. Divide x into +3x

b. Multiply +3 times (x+2)

c. Subtract like terms

Final Answer is: (x+3)

Try this: Divide (x³+8) by (x+2)

Synthetic Division

Synthetic division is a short method of polynomial division. The following example will explain
better about synthetic division.

Example: Divide x² + 5x + 6 by (x-1)

1. For the divisor, write
(x-1) = 0
x-1 = 0
x = 1

2. Write the coefficients of the dividend x² + 5x + 6 in the following manner

3. Now write the value x =1 from step1 to the left.

4. Carry down the first number from the left (leading coefficient).

5. Multiply this number by 1 (value of x written on the left) and write the result on the next column.

6. Add down the column.

7. Multiply the result by 1 (value of x on the left) and write the result in the next column.

8. Add down the column.

This should be recognized as (1x + 6) which is our quotient and the number 12 to the right is always the remainder.
Final Answer: Quotient = (x + 6)
Remainder = 12

Try this: Divide 2x² + 3x -8 by (x-3)