## Polynomials

1. Terms2. Polynomials

3. Evaluate Polynomials

4. Add/ Subtract Polynomials

5. Multiply Polynomials

6. Divide a Polynomial by a Monomial

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### 1. Terms

Terms are parts of an expression separated by plus or minus signs. A term is a product of

♦ a coefficient, which is the number in front of the term the plus/minus in front of the term belongs to the coefficient

♦ a variable part.

Example. The expression − 3

*x*

^{2}+ 2

*x*− 7 sin

*x*− 4 ln

*x*+ 8 has 5 terms:

• first term is − 3

*x*

^{2}, coefficient is − 3 and variable part is

*x*

^{2}

• second term is + 2

*x*, coefficient is 2 and variable part is

*x*

• third term is − 7 sin

*x*, coefficient is − 7 and variable part is sin

*x*

• fourth term is − 4 ln

*x*, coefficient is − 4 and variable part is ln

*x*

• fifth term is

**constant term**8

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### 2. Polynomials

Polynomials are sums of terms which are of the form

*a*where

_{n}x^{n}•

*a*are coefficients

_{n}•

*x*are variable parts

^{n}•

*n*≥ 0 are non-negative integers

Example. The expression 4

*x*

^{5}+

*x*

^{4}+ 2

*x*

^{3}−

*x*+ 8 is a polynomial of

__degree__5 written in

__descending order__.

•

*a*

_{5}= 4 •

*a*

_{4}= 1 •

*a*

_{3}= 2 •

*a*

_{2}= 0 •

*a*

_{1}= − 1 •

*a*

_{0}= 8

The

__leading coefficient__of this polynomial is

*a*

_{5}= 4 .

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### 3. Evaluate Polynomials

To evaluate an expression (in our case, a polynomial) is to replace the variable with its given value, then use order of operations to reduce it to a single value.Example. Evaluate 4

*x*

^{3}+ 7

*x*

^{2}−

*x*+ 8 for x = 3.

Answer. 4 ( 3 )

^{3}+ 7 ( 3 )

^{2}− ( 3 ) + 8

= 4 ( 27 ) + 7 ( 9 ) − ( 3 ) + 8

= 108 + 63 − 3 + 8

= 176

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### 4. Add/ Subtract Polynomials

**Reminder.**

• Two terms are

__like__if their variable parts are identical, for example 2

*x*

^{3}and 7

*x*

^{3}

• Two or more like terms can be added together or subtracted from one another by

† adding/subtracting the coefficiemts

† leaving the variable parts unchanged

2

*x*

^{3}+ 7

*x*

^{3}can be combined to be 9

*x*

^{3}

2

*x*

^{3}− 7

*x*

^{3}can be combined to be − 5

*x*

^{3}

Adding polynomials is adding their like terms.

Example. ( 4

*x*

^{3}+ 7

*x*

^{2}−

*x*+ 8 ) + ( − 4

*x*

^{3}+

*x*

^{2}+ 3

*x*) = 8

*x*

^{2}+ 2

*x*+ 8

Subtracting polynomials is subtracting their like terms.

To subtract one polynomial from the other:

1. Change subtraction to addition

2. Change the signs of all terms in the second polynomial (subtrahend)

3. Add

Example. ( 4

*x*

^{3}+ 7

*x*

^{2}−

*x*+ 8 ) − ( − 4

*x*

^{3}+

*x*

^{2}+ 3

*x*)

= ( 4

*x*

^{3}+ 7

*x*

^{2}−

*x*+ 8 ) + ( 4

*x*

^{3}−

*x*

^{2}− 3

*x*)

= 8

*x*

^{3}+ 6

*x*

^{2}− 4

*x*+ 8

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### 5. Multiply Polynomials

**Reminder.**

• To multiply two terms, we multiply their coefficients then add the exponents of same variables

• Distributive rule:

*a*(

*b*+

*c*) =

*a b*+

*a c*

Example. ( 2

*x*

^{3}

*y*) ( 7

*x*

^{2}

*y*

^{3}) can be combined to be 14

*x*

^{5}

*y*

^{4}

To multiply two polynomials:

1. Multiply each term of the first polynomial to every term of the second polynomial.

2. Combine like terms.

Examples.

• 5 ( 3

*x*

^{2}+

*x*- 3 ) = 15

*x*

^{2}+ 5

*x*- 15

• − 2

*x*( 3

*x*

^{2}+

*x*− 3 ) = − 6

*x*

^{3}− 2

*x*

^{2}+ 6

*x*

• ( − 2

*x*+ 7 ) ( 3

*x*

^{2}+

*x*− 3 ) = − 6

*x*

^{3}− 2

*x*

^{2}+ 6

*x*+ 21

*x*

^{2}+ 7

*x*− 21

= − 6

*x*

^{3}+ 19

*x*

^{2}+ 13

*x*− 21

Square of binomials: (

*a*+

*b*)

^{2}= (

*a*+

*b*) (

*a*+

*b*) =

*a*

^{2}+ 2

*ab*+

*b*

^{2}

1. Square each term

2. Multiply the two terms, double the coefficient, include this in the result

Example.

• ( 3

*x*+ 2 )

^{2}= 9

*x*

^{2}+ 12

*x*+ 4

• ( 2

*x*

^{2}− 5

*x*)

^{2}= 4

*x*

^{4}− 20

*x*

^{3}+ 25

*x*

^{2}

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### 6. Divide a Polynomial by a Monomial

• Use Distributive rule: divide each term of the polynomial by the monomial