Published On:Thursday, 8 December 2011
Posted by Muhammad Atif Saeed
Monomials, Polynomials, and Factoring
Monomials
A monomial is an algebraic expression that consists of only one term. (A term is a numerical or literal expression with its own sign.) For instance, 9 x, 4 a2, and 3 mpx2 are all monomials. The number in front of the variable is called the numerical coefficient. In 9 x, 9 is the coefficient.Adding and subtracting monomials
To add or subtract monomials, follow the same rules as with signed numbers, provided that the terms are alike. Notice that you add or subtract the coefficients only and leave the variables the same.Example 1
Perform the operation indicated.-
- 3 x + 2 x = 5 x
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Multiplying monomials
Reminder: The rules and definitions for powers and exponents also apply in algebra.
To multiply monomials, add the exponents of the same bases.
Example 2
Multiply the following.- ( x3)( x4) = x3 + 4 = x7
- ( x2 y)( x3 y2) = ( x2 x3)( yy2) = x2 + 3 y1 + 2 = x5 y3
- (6 k5)(5 k2) = (6 × 5)( k5 k2) = 30 k5 + 2 = 30 k7 (multiply numbers)
- –4( m2 n)(–3 m4 n3) = [(–4)(–3)]( m2 m4)( nn3) = 12 m2 + 4 n1 + 3 = 12 m6 n4 (multiply numbers)
- ( c2)( c3)( c4) = c2 + 3 + 4 = c9
- (3 a2 b3 c)( b2 c2 d) = 3( a2)( b3 b2)( cc2)( d) = 3 a2 b3 + 2 c1 + 2 d = 3 a2 b5 c3 d
When monomials are being raised to a power, the answer is obtained by multiplying the exponents of each part of the monomial by the power to which it is being raised.
Example 3
Simplify.- ( a7)3 = a21
- ( x3 y2)4 = x12 y8
- (2 x2 y3)3 = (2)3 x6 y9 = 8 x6 y9
Dividing monomials
To divide monomials, subtract the exponent of the divisor from the exponent of the dividend of the same base.Example 4
Divide.
Working with negative exponents
Remember, if the exponent is negative, such as x–3, then the variable and exponent may be dropped under the number 1 in a fraction to remove the negative sign as follows.
Example 5
Express the answers with positive exponents.
Polynomials
A polynomial consists of two or more terms. For example, x + y, y2 – x2, and x2 + 3 x + 5 y2 are all polynomials. A binomial is a polynomial that consists of exactly two terms. For example, x + y is a binomial. A trinomial is a polynomial that consists of exactly three terms. For example, y2 + 9 y + 8 is a trinomial.Polynomials usually are arranged in one of two ways. Ascending order is basically when the power of a term increases for each succeeding term. For example, x + x2 + x3 or 5 x + 2 x2 – 3 x3 + x5 are arranged in ascending order. Descending order is basically when the power of a term decreases for each succeeding term. For example, x3 + x2 + x or 2 x4 + 3 x2 + 7 x are arranged in descending order. Descending order is more commonly used.
Adding and subtracting polynomials
To add or subtract polynomials, just arrange like terms in columns and then add or subtract. (Or simply add or subtract like terms when rearrangement is not necessary.)Example 1
Do the indicated arithmetic.- Add the polynomials.
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- Subtract the polynomials.
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Multiplying polynomials
To multiply polynomials, multiply each term in one polynomial by each term in the other polynomial. Then simplify if necessary.Example 2
Multiply.
Example 3
Multiply.(3 x + a)(2 x – 2 a) =
Multiply first terms from each quantity. 
6 x2 – 6 ax + 2 ax – 2 a2 = 6 x2 – 4 ax – 2 a2
Example 4
Multiply.
Dividing polynomials by monomials
To divide a polynomial by a monomial, just divide each term in the polynomial by the monomial.Example 5
Divide.
Dividing polynomials by polynomials
To divide a polynomial by a polynomial, make sure both are in descending order; then use long division. ( Remember: Divide by the first term, multiply, subtract, bring down.)Example 6
Divide 4 a2 + 18 a + 8 by a + 4.
Example 7
Divide.-
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First change to descending order: x2 + 2 x + 1. Then divide.
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Note: When terms are missing, be sure to leave proper room between terms.
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Factoring
To factor means to find two or more quantities whose product equals the original quantity.Factoring out a common factor
To factor out a common factor, (1) find the largest common monomial factor of each term and (2) divide the original polynomial by this factor to obtain the second factor. The second factor will be a polynomial.Example 1
Factor.- 5 x2 + 4 x = x(5 x + 4)
- 2 y3 – 6 y = 2 y( y2 – 3)
- x5 – 4 x3 + x2 = x2( x3 – 4 x + 1)
Factoring the difference between two squares
To factor the difference between two squares, (1) find the square root of the first term and the square root of the second term and (2) express your answer as the product of the sum of the quantities from Step 1 times the difference of those quantities.Example 2
Factor.- x2 – 144 = ( x + 12)( x – 12)
Note: x2 + 144 is not factorable. - a2 – b2 = ( a + b)( a – b)
- 9 y2 – 1 = (3 y + 1)(3 y – 1)
Factoring polynomials having three terms of the form ax 2 + bx + c
To factor polynomials having three terms of the form ax2 + bx + c, (1) check to see whether you can monomial factor (factor out common terms). Then if a = 1 (that is, the first term is simply x2), use double parentheses and factor the first term. Place these factors in the left sides of the parentheses. For example,( x )( x )
(2) Factor the last term and place the factors in the right sides of the parentheses.To decide on the signs of the numbers, do the following. If the sign of the last term is negative, (1) find two numbers (one will be a positive number and the other a negative number) whose product is the last term and whose difference is the coefficient (number in front) of the middle term and (2) give the larger of these two numbers the sign of the middle term and the opposite sign to the other factor.
If the sign of the last term is positive, (1) find two numbers (both will be positive or both will be negative) whose product is the last term and whose sum is the coefficient of the middle term and (2) give both factors the sign of the middle term.
Example 3
Factor x2 – 3 x – 10.First check to see whether you can monomial factor (factor out common terms). Because this is not possible, use double parentheses and factor the first term as follows: ( x ) ( x ). Next, factor the last term, 10, into 2 times 5 (5 must take the negative sign and 2 must take the positive sign because they will then total the coefficient of the middle term, which is –3) and add the proper signs, leaving
( x – 5)( x + 2)
Multiply means (inner terms) and extremes (outer terms) to check. 
Example 4
Factor x2 + 8 x + 15.( x + 3)( x + 5)
Notice that 3 × 5 = 15 and 3 + 5 = 8, the coefficient of the middle term. Also note that the signs of both factors are +, the sign of the middle term. To check, 
Example 5
Factor x2 – 5 x – 14.( x – 7)( x + 2)
Notice that 7 × 2 = 14 and 7 – 2 = 5, the coefficient of the middle term. Also note that the sign of the larger factor, 7, is –, while the other factor, 2, has a + sign. To check, 
Example 6
Factor 4 x2 +5 x + 1.(2 x + )(2 x + ) might work for the first term. But when 1s are used as factors to get the last term, (2 x + 1)(2 x + 1), the middle term comes out as 4 x instead of 5 x.
Example 7
Factor 4 a2 + 6 a + 2.Factoring out a 2 leaves
2(2 a2 + 3 a + 1)
Now factor as usual, giving2(2 a + 1)( a + 1)
To check, 
Example 8
Factor 5 x3 + 6 x2 + x.Factoring out an x leaves
x(5 x2 + 6 x + 1)
Now factor as usual, giving x(5 x + 1)( x + 1)
To check, 
Example 9
Factor 5 + 7 b + 2 b2 (a slight twist).(5 + 2 b)(1 + b)
To check, 
Example 10
Factor x2 + 2 xy + y2.( x + y)( x + y)
To check, 
Example 11
Factor 3 x2 – 48.Factoring out a 3 leaves
3( x2 – 16)
But x2 – 16 is the difference between two squares and can be further factored into ( x + 4)( x – 4). Therefore, when completely factored, 3 x2 – 48 = 3( x + 4)( x – 4). Factoring by grouping
Some polynomials have binomial, trinomial, and other polynomial factors.Example 12
Factor x + 2 + xy + 2 y.Since there is no monomial factor, you should attempt rearranging the terms and looking for binomial factors.
x + 2 + xy + 2 y = x + xy + 2 + 2 y
Grouping gives( x + xy) + (2 + 2 y)
Now factoring gives x(1 + y) + 2(1 + y)
Using the distributive property gives( x + 2)(1 + y)
You could rearrange them differently, but you would still come up with the same factoring.Summary of the factoring methods
When factoring polynomials, you should look for factoring in the following order.- Look for the greatest common factor if one exists.
- If there are two terms, look for the difference of square numbers.
- If there are three terms, look for a pattern that applies to ax2 + bx + c.
- If there are four or more terms, look for some type of regrouping that will lead to other factoring.
Example 13
Factor 2 x2 + 3 x + 5.- This polynomial does not have a common factor.
- This polynomial is not a difference of square numbers.
- There is no (_ x )(_ x ) combination that produces 2 x2 + 3 x + 5.
Since there are only three terms, there is no regrouping possibility.
Therefore, this polynomial is not factorable.
