POLYNOMIAL

Polynomials

 Definition:

A polynomial is a monomial or the sum or difference of monomials. Each monomial is called a term of the poynomial

Terms

Ascending Order  -  An ordering of terms of a polynomial in which the degree of each term is at least as large as the degree of the preceding term.

Binomial  -  A polynomial with two terms.

Common Factor  -  A factor of all the terms in an expression.

Degree  -  of a monomial. The sum of the exponents of its variables. of a polynomial. The largest of the degrees of its individual terms.

Descending Order  -  An ordering of terms of a polynomial in which the degree of each term is no larger than the degree of the preceding term.

Difference of Squares  -  The binomial which is the product of the sum and difference of the same two terms, i.e. one of the form a 2 - b 2 .

Factor  -  A factor is integer, variable, or polynomial that can be multiplied by a constant, an integer, or a polynomial to produce the given expression. To factor means to separate an expression into simpler factors. For instance,

a 2 -2ab + b 2      =             (a - b)(a - b)       

ax 2 + abx + ac   =             a(x 2 + bx + c)   

Leading Coefficient  -  The coefficient of the first term of a polynomial written in descending order.

Monomial  -  An expression with a single term; a real number, a variable, or the product of real numbers and variables

Perfect Square Trinomial  -  The square of a binomial; has the form a 2 +2ab + b 2 .

Polynomial  -  A monomial or the sum or difference of several monomials.

Trinomial  -  A polynomial with three terms.

Classification of Polynomials

Polynomials are classified according to two attributes -- number of terms and degree.

Classification of Polynomials by Number of Terms

A monomial is an expression with a single term. It is a real number, a variable, or the product of real numbers and variables. For example, 4 , 3x 2 , and 15xy 3 are all monomials, but 4x 2 + x , (3 + y)2 , and 12 - z are not monomials.

A polynomial is a monomial or the sum or difference of monomials. 4x 3 +3y + 3x 2 + z , -12zy , and 15 - x 2 are all polynomials.

Polynomials are classified according to their number of terms. 4x 3 +3y + 3x 2 has three terms, -12zy has 1 term, and 15 - x 2 has two terms. As already mentioned, a polynomial with 1 term is a monomial. A polynomial with two terms is a binomial, and a polynomial with three terms is a trinomial.

Classification of Polynomials by Degree

The degree of a monomial is the sum of the exponents of its variables. For example, 12x 3 has degree 3, x 2 y 5 has degree 2 + 5 = 7 , and 11xy has degree 1 + 1 = 2 .

A polynomial can be arranged in ascending order, in which the degree of each term is at least as large as the degree of the preceding term, or in descending order, in which the degree of each term is no larger than the degree of the preceding term. The polynomial 3 + 12x - xy + 7x 2 y + y 5 -12x 3 y 3 is written in ascending order, while the same polynomial expressed as -12x 3 y 3 + y 5 +7x 2 y - xy + 12x + 3 is written in descending order.

Mathematicians generally write polynomials in descending order. The coefficient of the first term of a polynomial written in descending order is known as the leading coefficient.

The degree of a polynomial is the largest of the degrees of its monomial terms.

Addition and Subtraction of Polynomials

Addition of Polynomials

To add two polynomials, combine their like terms.

Example 1: (4x 5 + x 4 -12x 3 + x - 6) + (3x 4 +8x 3 +6x 2 - x) = ?

= 4x 5 + (1 + 3)x 4 + (- 12 + 8)x 3 +6x 2 + (- 1 + 1)x - 6

= 4x 5 +4x 4 -4x 3 +6x 2 + 0x - 6

= 4x 5 +4x 4 -4x 3 +6x 2 - 6

Example 2: (- 3x 4 -2x 2 + x + 17) + (14x 6 +12x 5 -2x 4 +7x 3 +2x 2 - 4x - 12) = ?

= 14x 6 +12x 5 + (- 3 - 2)x 4 +7x 3 + (- 2 + 2)x 2 + (1 - 4)x + (17 - 12)

= 14x 6 +12x 5 -5x 4 +7x 3 +0x 2 - 3x + 5

= 14x 6 +12x 5 -5x 4 +7x 3 - 3x + 5

Subtraction of Polynomials

To subtract one polynomial from another, change the subtraction sign to an addition sign and change the signs of all the terms in the polynomial being subtracted (don't forget to change the sign of the first term). Then add the resulting polynomial by combining like terms.

Example 3: (3x 4 +4x 3 -5x 2 +18) - (7x 5 +3x 4 -12x 3 -3x 2 + 7x - 9) = ?

= (3x 4 +4x 3 -5x 2 +18) + (- 7x 5 -3x 4 +12x 3 +3x 2 - 7x + 9)

= - 7x 5 + (3 - 3)x 4 + (4 + 12)x 3 + (- 5 + 3)x 2 - 7x + (18 + 9)

= - 7x 5 +16x 3 -2x 2 - 7x + 27

Example 4: (- 5x 6 -3x 4 +18x 3 -7x 2 +9x + 11) - (- x 4 + x 3 -12x 2 + 10x + 5) = ?

= (- 5x 6 -3x 4 +18x 3 -7x 2 +9x + 11) + (x 4 - x 3 +12x 2 - 10x - 5)

= - 5x 6 + (- 3 + 1)x 4 + (18 - 1)x 3 + (- 7 + 12)x 2 + (9 - 10)x + (11 - 5)

= - 5x 6 -2x 4 +17x 3 +5x 2 - x + 6

Multiplication of Polynomials

Multiplication of a Polynomial by a Monomial

To multiply a polynomial by a monomial, use the distributive property: multiply each term of the polynomial by the monomial. This involves multiplying coefficients and adding exponents of the appropriate variables.

Example 1: 3y 2(12y 3 -6y 2 + 5y - 1) = ?

= 3y 2(12y 3) + (3y 2)(- 6y 2) + (3y 2)(5y) + (3y 2)(- 1)

= (3)(12)y 2+3 + (3)(- 6)y 2+2 + (3)(5)y 2+1 + (3)(- 1)y 2

= 36y 5 -18y 4 +15y 3 -3y 2

Example 2: -4x 3 y(- 2y 2 + xy - x + 9) = ?

= - 4x 3 y(- 2y 2) + (- 4x 3 y)(xy) + (- 4x 3 y)(- x) + (- 4x 3 y)(9)

= (- 4)(- 2)x 3 y 1+2 + (- 4)x 3+1 y 1+1 + (- 4)(- 1)x 3+1 y + (- 4)(9)x 3 y

= 8x 3 y 3 -4x 4 y 2 +4x 4 y - 36x 3 y

Multiplication of Binomials

To multiply a binomial by a binomial-- (a + b)(c + d ) , where a , b , c , and d are terms--use the distributive property twice. First, treat the second binomial as a single term and distribute over the first binomial:

(a + b)(c + d )= a(c + d )+ b(c + d )                 

Next, use the distributive property over the second binomial:

a(c + d )+ b(c + d )= ac + ad + bc + bd          

At this point, there should be 4 terms in the answer -- every combination of a term of the first binomial and a term of the second binomial. Simplify the answer by combining like terms.

We can use the word FOIL to remember how to multiply two binomials (a + b)(c + d ) :

Multiply their First terms. (ac)

Multiply their Outside terms. (ad )

Multiply their Iinside terms. (bc)

Multiply their Last terms. (bd )

Finally, add the results together: ac + ad + bc + bd . Combine like terms.

Remember to include negative signs as part of their respective terms in the multiplication.

Example 1. (xy + 6)(x + 2y) = ?

= (xy)(x) + (xy)(2y) + (6)(x) + (6)(2y)

= x 2 y + 2xy 2 + 6x + 12y

Example 2. (3x 2 +7)(4 - x 2) = ?

= (3x 2)(4) + (3x 2)(- x 2) + (7)(4) + (7)(- x 2)

= 12x 2 -3x 4 +28 - 7x 2

= - 3x 4 + (12 - 7)x 2 + 28

= - 3x 4 +5x 2 + 28

Example 3: (y - x)(- 4y - 3x) = ?

= (y)(- 4y) + (y)(- 3x) + (- x)(- 4y) + (- x)(- 3x)

= - 4y 2 -3xy + 4xy + 3x 2

= 3x 2 + (- 3 + 4)xy - 4y 2

= 3x 2 + xy - 4y 2

Multiplication of Polynomials

The strategy for multiplying two polynomials in general is similar to multiplying two binomials. First, treat the second polynomial as a single term, and distribute over the first term:

(a + b + c)(d + e + f )= a(d + e + f )+ b(d + e + f )+ c(d + e + f )           

Next, distribute over the second polynomial:

a(d + e + f )+ b(d + e + f )+ c(d + e + f )= ad + ae + af + bd + be + bf + cd + ce + cf                    

At this point, the number of terms in the answer should be the number in the first polynomial times the number in the second polynomial--every combination of a term of the first polynomial and a term of the second polynomial. Since there are 3 terms in each polynomial in this example there should be 3(3) = 9 terms in our answer so far. If the first polynomial had 4 terms and the second had 5 , there would be 4(5) = 20 terms in the answer so far.

Finally, since the the terms in such a product of polynomials are often highly redundant (many have the same variables and exponents), it is important to combine like terms.

Example 1: (x 2 -2)(3x 2 - 3x + 7) = ?

= x 2(3x 2 -3x + 7) - 2(3x 2 - 3x + 7)

= x 2(3x 2) + x 2(- 3x) + x 2(7) - 2(3x 2) - 2(- 3x) - 2(7) ( 6 terms)

= 3x 4 -3x 3 +7x 2 -6x 2 + 6x - 14

= 3x 4 -3x 3 + (7 - 6)x 2 + 6x - 14

= 3x 4 -3x 3 + x 2 + 6x - 14

Example 2: (x 2 + x + 3)(2x 2 - 3x + 1) = ?

= x 2(2x 2 -3x + 1) + x(2x 2 -3x + 1) + 3(2x 2 - 3x + 1)

= x 2(2x 2) + x 2(- 3x) + x 2(1) + x(2x 2) + x(- 3x) + x(1) + 3(2x 2) + 3(- 3x) + 3(1) ( 9 terms)

= 2x 4 -3x 3 + x 2 +2x 3 -3x 2 + x + 6x 2 - 9x + 3

= 2x 4 + (- 3 + 2)x 3 + (1 - 3 + 6)x 2 + (1 - 9)x + 3

= 2x 4 - x 3 +4x 2 - 8x + 3

Note: To check your answer, pick a value for the variable and evaluate both the original expression and your answer--they should be the same.

Multiplication of Binomials -- Special Cases

Square of a Binomial

To square a binomial, multiply the binomial by itself:

(a + b)2 = (a + b)(a + b)

(a + b)2 =             (a + b)(a + b)     

                =             a 2 + ab + ba + b 2             

                =             a 2 + ab + ab + b 2             

                =             a 2 +2ab + b 2      

The square of a binomial is always the sum of:

The first term squared,

2 times the product of the first and second terms, and

the second term squared.

When a binomial is squared, the resulting trinomial is called a perfect square trinomial.

Examples:

(x + 5)2 = x 2 +2(x)(5) + 52 = x 2 + 10x + 25

(100 - 1)2 = 1002 +2(100)(- 1) + (- 1)2 = 10000 - 200 + 1 = 9801

(2x - 3y)2 = (2x)2 +2(2x)(- 3y) + (- 3y)2 = 4x 2 -12xy + 9y 2

Product of the Sum and Difference of Two Terms

When we multiply two polynomials that are the sum and difference of the same 2 terms -- (x + 5) and (x - 5) for example -- we get an interesting result:

(a + b)(a - b)   = a(a) + a(- b) + ba + b(- b)              

                =             a 2 - ab + ab - b 2               

                =             a 2 - b 2                  

The product of the sum and difference of the same two terms is always the difference of two squares; it is the first term squared minus the second term squared. Thus, this resulting binomial is called a difference of squares.

Examples:

(7 - 2)(7 + 2) = 72 -22 = 49 - 4 = 45

(x + 9)(x - 9) = x 2 -92 = x 2 - 81

(2x - y)(2x + y) = (2x)2 - y 2 = 4x 2 - y 2

(3x 2 -2)(3x 2 +2) = (3x 2)2 -22 = 9x 4 - 4

(- y + 5x)(- y - 5x) = (- y)2 - (5x)2 = y 2 -15x 2

Removing Common Factors

Factors

A factor is a number that evenly divides the given number. A factor need not be a constant. In fact, any integer, variable, or polynomial that can be multiplied by an integer, a variable, or a polynomial to produce the given expression is a factor of the given expression.

Removing Common Factors

We've seen how to distribute a quantity over a polynomial and write the result as a polynomial. We can actually reverse this process--we can "remove" a common factor from a polynomial and write the result as a quantity times a polynomial. For example, 12 + 2x can be written as 2(6 + x) .

The first step to removing a common factor is finding a common factor. A common factor is a factor of all the terms in an expression (i.e., a factor that they all have in common). A common factor can be an integer, a variable, or a combination of integers and variables.

To remove a common factor and rewrite a polynomial as the product of a monomial and another polynomial:

Find the greatest common factor which is a whole number (no variables).

Divide all terms of the polynomial by that factor, and put the result in parentheses. Write the factor outside the parentheses.

Find the greatest common factor which is a variable or a product of several variables. That is, find the variables contained in every term, and write them with their lowest exponent.

Divide each term of the expression in parentheses by the greatest common variable factor, and write the variable factor outside the parentheses.

Check--distributing the monomial over the new polynomial should yield the original polynomial.

Example 1: Factor 4x 2 +16x 3 + 8x .

The greatest common whole number factor is 4 .

4x 2 +16x 3 +8x = 4(x 2 +4x 3 + 2x)

The greatest common variable factor is x ( x is contained in all the terms, and its lowest exponent is 1 ).

4(x 2 +4x 3 +2x) = 4x(x + 4x 2 + 2)

Check: 4x(x + 4x 2 +2) = 4x 2 +16x 3 + 8x

Thus, 4x 2 +16x 3 +8x = 4x(x + 4x 2 + 2) .

Example 2: Factor 12x 3 y + 3x 4 y 2 -6x 2 y 2 z .

The greatest common whole number factor is 3 .

12x 3 y + 3x 4 y 2 -6x 2 y 2 z = 3(4x 3 y + x 4 y 2 -2x 2 y 2 z)

The greatest common variable factor is x 2 y ( x is contained in all the terms, and its lowest exponent is 2 ; y is contained in all the terms, and its lowest exponent is 1; z is not contained in all the terms).

3(4x 3 y + x 4 y 2 -2x 2 y 2 z) = 3x 2 y(4x + x 2 y - 2yz)

Check: 3x 2 y(4x + x 2 y - 2yz) = 12x 3 y + 3x 4 y 2 -6x 2 y 2 z

Thus, 12x 3 y + 3x 4 y 2 -6x 2 y 2 z = 3x 2 y(4x + x 2 y - 2yz) .

Factoring Trinomials

Factoring Trinomials of the Form x 2 + bx + c and x 2 - bx + c

Just as the product of two binomials can often be rewritten as a trinomial, trinomials of the form ax 2 + bx + c can often be rewritten as the product of two binomials. For example, x 2 + 3x + 2 = (x + 1)(x + 2) .

We now know that the product of two binomials of the form (x + d ) and (x + e) is given by:

(x + d )(x + e) = x 2 + xe + dx + de = x 2 + (d + e)x + de        

Thus, in order to rewrite a binomial x 2 + bx + c as the product of two binomials ( b positive or negative), we must find numbers d and e such that d + e = b and de = c . Since c is positive, d and e must have the same sign.

Here are the steps to factoring a trinomial of the form x 2 + bx + c , with c > 0 . We assume that the coefficients are integers, and that we want to factor into binomials with integer coefficients.

Write out all the pairs of numbers which can be multiplied to produce c .

Add each pair of numbers to find a pair that produce b when added. Call the numbers in this pair d and e .

If b > 0 , then the factored form of the trinomial is (x + d )(x + e) . If b < 0 , then the factored form of the trinomial is (x - d )(x - e) .

Check: The binomials, when multiplied, should equal the original trinomial.

Note: Some trinomials cannot be factored. If none of the pairs total b , then the trinomial cannot be factored.

Example 1: Factor x 2 + 5x + 6 .

Pairs of numbers which make 6 when multiplied: (1, 6) and (2, 3) .

1 + 6≠5 . 2 + 3 = 5. Thus, d = 2 and e = 3 (or vice versa).

(x + 2)(x + 3)

Check: (x + 2)(x + 3) = x 2 +3x + 2x + 6 = x 2 + 5x + 6

Thus, x 2 + 5x + 6 = (x + 2)(x + 3) .

Example 2: Factor x 2 - 7x + 12 .

Pairs of numbers which make 12 when multiplied: (1, 12) , (2, 6) , and (3, 4) .

1 + 12≠7 . 2 + 6≠7 . 3 + 4 = 7. Thus, d = 3 and e = 4 .

(x - 3)(x - 4)

Check: (x - 3)(x - 4) = x 2 -4x - 3x + 12 = x 2 - 7x + 12

Thus, x 2 - 7x + 12 = (x - 3)(x - 4) .

Example 3: Factor 2x 3 +4x 2 + 2x .

First, remove common factors: 2x 3 +4x 2 +2x = 2x(x 2 + 2x + 1)

Pairs of numbers which make 1 when multiplied: (1, 1) .

1 + 1 = 2. Thus, d = 1 and e = 1 .

2x(x + 1)(x + 1) (don't forget the common factor!)

Check: 2x(x + 1)(x + 1) = 2x(x 2 +2x + 1) = 2x 3 +4x 2 + 2x

Thus, 2x 3 +4x 2 +2x = 2x(x + 1)(x + 1) = 2x(x + 1)2 .

x 2 + 2x + 1 is a perfect square trinomial.

Factoring Trinomials of the Form x 2 + bx - c and x 2 - bx - c

In the equation (x + d )(x + e) = x 2 + (d + e)x + de = x 2 + bx + c (mentioned in Heading ), c is negative if and only if de is negative; that is, if and only if d is negative or e is negative. Thus, the equation becomes (x + d )(x - e) = x 2 + (d - e)x - de = x 2 + bx - c , which c > 0 .

In order to rewrite the trinomial x 2 + bx - c as the product of two binomials, we must find numbers d and e such that d - e = b and | de| = c . If b is positive, then d - e > 0 , so d > e . If b is negative, then d - e < 0 , so d < e .

Here are the steps to factoring a trinomial of the form x 2 + bx - c , with c > 0 :

Write out all the pairs of numbers which can be multiplied to produce c .

Subtract each pair of numbers to find a pair that produce b when one is subtracted from the other. Call the numbers in this pair d and e . If b > 0 , let d be the larger number, and if b < 0 , let d be the smaller number.

The factored form of the trinomial is (x + d )(x - e) .

Check: The binomials, when multiplied, should equal the original trinomial. If the middle term has the wrong sign, you most likely switched d and e . Switch the "+" and "-" sign in your binomials and check again.

Note: Again, not every trinomial can be factored.

Example 1: Factor x 2 + 6x - 16 .

Pairs of numbers which make 16 when multiplied: (1, 16) , (2, 8) , and (4, 4) .

16 - 1≠6 . 8 - 2 = 6. Since b = 6 > 0 , d = 8 and e = 2 .

(x + 8)(x - 2)

Check: (x + 8)(x - 2) = x 2 -2x + 8x - 16 = x 2 + 6x - 16

Thus, x 2 + 6x - 16 = (x + 8)(x - 2) .

Example 2: Factor x 2 - x - 20 .

Pairs of numbers which make 20 when multiplied: (1, 20) , (2, 10) , and (4, 5) .

20 - 1≠1 . 10 - 2≠1 . 5 - 4 = 1. Since b = - 1 < 0 , d = 4 and e = 5 .

(x + 4)(x - 5)

Check: (x + 4)(x - 5) = x 2 -5x + 4x - 20 = x 2 - x - 20

Thus, x 2 - x - 20 = (x + 4)(x - 5) .

Example 3: Factor x 2 - 16 (Here, x has an implied coefficient of b = 0 ).

Pairs of numbers which make 16 when multiplied: (1, 16) , (2, 8) , and (4, 4) .

16 - 1≠ 0 . 8 - 2≠ 0 . 4 - 4 = 0. Thus, d = 4 and e = 4 .

(x + 4)(x - 4)

Check: (x + 4)(x - 4) = x 2 -4x + 4x - 16 = x 2 - 16

Thus, x 2 - 16 = (x + 4)(x - 4) .