Polynomials

• Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms“

Polynomials

• A polynomial looks like this:

Term

A number, a variable, or the product/quotient of

numbers/variables.

A term has 3 components:

Coefficient: can by any real

number… including zero.

Variable:

Usually denoted as x

Exponent:Can only be 0,1,2, 3, …

A polynomial can have:

constants (like 3, -20, or ½)

variables (like x and y)

exponents (like the 2 in y2), but only 0, 1, 2, 3, ... etc are allowed

Remember

These are more examples of polynomials

3x x - 2 -6y2 - (7/9)x

3xyz + 3xy2z - 0.1xz - 200y + 0.5 512v5+ 99w5 5 (which is really 5x0)

• A polynomial can have: Terms that can be combined by adding, subtracting or multiplying.

• A Polynomial can NOT have a division by a variable in it.

These are not allowed as a term in a polynomial

• is not, because dividing by a variable is not allowed

• is not either• 3xy-2 is not, because : • is not, because the exponent is "½"

These are allowed as a term in a polynomial

• is allowed, because you can divide by a constant

• also for the same reason• is allowed, because it is a constant (=

1.4142...etc)

Monomial, binomial, and trinomial

• There is also quadrinomial (4 terms) and quintinomial (5 terms), but those names are not often used.

• Can Have Lots and Lots of Terms• Polynomials can have as many terms as

needed, but not an infinite number of terms.

Standard Form

• The Standard Form for writing a polynomial is to put the terms with the highest degree first.

• Example: Put this in Standard Form: 3x2 - 7 + 4x3 + x6

• The highest degree is 6, so that goes first, then 3, 2 and then the constant last:

• x6 + 4x3 + 3x2 - 7

General Term of a polynomial

•  

Degree of a Term

The exponent of the variable.

We will find them only for one-variable terms.

Term

3

4x

-5x2

18x5

Degree of Term

0

1

2

5

PolynomialA term or the

sum/difference of terms which contain only 1

variable. The variable cannot be in the

denominator of a term.

Degree of a Polynomial

The degree of the term with the highest degree.

Polynomial

6x2 - 3x + 2

15 - 4x + 5x4

x + 10 + x

x3 + x2 + x + 1

Degree ofPolynomial

2

4

1

3

Standard Form of a Polynomial

A polynomial written so that the degree of the

terms decreases from left to right and no terms have

the same degree.

Not Standard

6x + 3x2 - 2

15 - 4x + 5x4

x + 10 + x

1 + x2 + x + x3

Standard

3x2 + 6x - 2

5x4 - 4x + 15

2x + 10

x3 + x2 + x + 1

Naming Polynomials

Polynomials are named or classified by their

degree and the number of terms they

have.

Polynomial

7

5x + 2

4x2 + 3x - 4

6x3 - 18

Degree

0

1

2

3

Degree Name

constant

linear

quadratic

cubicFor degrees higher than 3 say:

“of degree n” or “nth degree”

x5 + 3x2 + 3 “of degree 5” – or – “5th degree”

x8 + 4 “of degree 8” – or – “8th degree”

Polynomial

7

5x + 2

4x2 + 3x - 4

6x3 - 18

# Terms

1

2

3

2

# Terms Name

monomial

binomial

trinomial

binomialFor more than 3 terms say:

“a polynomial with n terms” or “an n-term polynomial”

11x8 + x5 + x4 - 3x3 + 5x2 - 3“a polynomial with 6 terms” – or – “a 6-term

polynomial”

Polynomial

-14x3

-1.2x2

-1

7x - 2

3x3+ 2x - 8

2x2 - 4x + 8

x4 + 3

Name

cubic monomial

quadratic monomial

constant monomial

linear binomial

cubic trinomial

quadratic trinomial

4th degree binomial

Adding and Subtracting Polynomials

To add or subtract polynomials, simply

combine like terms.

(5x2 - 3x + 7) + (2x2 + 5x - 7)

= 7x2 + 2x

(3x3 + 6x - 8) + (4x2 + 2x - 5)

= 3x3 + 4x2 + 8x - 13

(2x3 + 4x2 - 6) – (3x3 + 2x - 2)

(7x3 - 3x + 1) – (x3 - 4x2 - 2)

(2x3 + 4x2 - 6) + (-3x3 + -2x - -2)

= -x3 + 4x2 - 2x - 4

(7x3 - 3x + 1) + (-x3 - -4x2 - -2)

= 6x3 + 4x2 - 3x + 3

7y2 – 3y + 4+ 8y2 + 3y – 4

2x3 – 5x2 + 3x – 1

– (8x3 – 8x2 + 4x + 3)

–6x3 + 3x2 – x – 4

15y2 + 0y + 0 =

15y2

TR

Y I

T T

HE

VER

TIC

AL W

AY

!

(7y3 +2y2 + 5y – 1) + (5y3 + 7y)

7y3 + 2y2 + 5y – 1

+ 5y3 + 0y2 + 7y + 012y3 + 2y2 + 12y – 1

(b4 – 6 + 5b + 1) + (8b4 + 2b – 3b2)

Rewrite in standard form!

b4 + 0b3 + 0b2 + 5b – 5+ 8b4 + 0b3 – 3b2 + 2b + 09b4 + 0b3 – 3b2 + 7b – 5

= 9b4 – 3b2 + 7b – 5

Polynomials: Algebra Tiles

Algebra tiles are tools that help one represent a polynomial.

Polynomials: Algebra Tiles

What polynomial is represented below?

Polynomials: Multiplying by a Monomial w/tiles

When you multiply, you are really finding the areas of rectangles. The length and width are the factors and the product is the area.

For example multiply 2x (3x + 1)

3x+1

2x

6x2 + 2x

MultiplyingPolynomials

Distribute and FOIL

Polynomials * Polynomials Polynomials * Polynomials

Multiplying a Polynomial by another Polynomial requires more than one distributing step.

Multiply: (2a + 7b)(3a + 5b)

Distribute 2a(3a + 5b) and distribute 7b(3a + 5b):

6a2 + 10ab 21ab + 35b2

Then add those products, adding like terms:

6a2 + 10ab + 21ab + 35b2 = 6a2 + 31ab + 35b2

Polynomials * Polynomials Polynomials * Polynomials

An alternative is to stack the polynomials and do long multiplication.

(2a + 7b)(3a + 5b)

6a2 + 10ab

21ab + 35b2

(2a + 7b)

x (3a + 5b)

Multiply by 5b, then by 3a:(2a + 7b)

x (3a + 5b)

When multiplying by 3a, line up the first term under 3a.

+

Add like terms: 6a2 + 31ab + 35b2

Polynomials * Polynomials Polynomials * Polynomials

Multiply the following polynomials:

1) x 5 2x 1

2) 3w 2 2w 5

3) 2a2 a 1 2a2 1

Polynomials * Polynomials Polynomials * Polynomials

1) x 5 2x 1 (x + 5)

x (2x + -1)

-x + -5

2x2 + 10x+

2x2 + 9x + -5

2) 3w 2 2w 5 (3w + -2)

x (2w + -5)

-15w + 10

6w2 + -4w+

6w2 + -19w + 10

Polynomials * Polynomials Polynomials * Polynomials

3) 2a2 a 1 2a2 1 (2a2 + a + -1)

x (2a2 + 1)

2a2 + a + -1

4a4 + 2a3 + -2a2+

4a4 + 2a3 + a + -1

Types of PolynomialsTypes of Polynomials

• We have names to classify polynomials based on how many terms they have:

Monomial: a polynomial with one term

Binomial: a polynomial with two terms

Trinomial: a polynomial with three terms

F : Multiply the First term in each binomial. 2x • 4x = 8x2

There is an acronym to help us remember how to multiply two binomials without stacking them.

F.O.I.L.F.O.I.L.

(2x + -3)(4x + 5)

(2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15

O : Multiply the Outer terms in the binomials. 2x • 5 = 10x

I : Multiply the Inner terms in the binomials. -3 • 4x = -12x

L : Multiply the Last term in each binomial. -3 • 5 = -15

Use the FOIL method to multiply these binomials:

F.O.I.L.F.O.I.L.

1) (3a + 4)(2a + 1)

2) (x + 4)(x - 5)

3) (x + 5)(x - 5)

4) (c - 3)(2c - 5)

5) (2w + 3)(2w - 3)

Use the FOIL method to multiply these binomials:

F.O.I.L.F.O.I.L.

1) (3a + 4)(2a + 1) = 6a2 + 3a + 8a + 4 = 6a2 + 11a + 4

2) (x + 4)(x - 5) = x2 + -5x + 4x + -20 = x2 + -1x + -20

3) (x + 5)(x - 5) = x2 + -5x + 5x + -25 = x2 + -25

4) (c - 3)(2c - 5) = 2c2 + -5c + -6c + 15 = 2c2 + -11c + 15

5) (2w + 3)(2w - 3) = 4w2 + -6w + 6w + -9 = 4w2 + -9