Polynomials

Algebra I

Vocabulary

• Monomial – a number, variable or a product of a number and one or more variables.

• Binomial – sum of two monomials.• Trinomial – sum of three monomials.• Polynomial – a monomial or a sum of

monomials.• Constants – monomials that are real

numbers.

Examples

• Monomialo 13no -5z

• Binomialo 2a + 3co 6x²+ 3xy

• Trinomialo p²+ 5p + 4o 3a²- 2ab - b²

More Vocabulary

• Terms – the individual monomial.• Degree of a monomial – sum of

the exponents of all it’s variables.• Degree of a polynomial – The

greatest degree of any term in the polynomial. To find the degree of a polynomial, you must first find the degree of each term.

Examples

• Monomial o 8y³ degree is 3o 3a degree is 1

• Polynomialo 5mn³degree is 4o -4x²y²+ 3x²y + 5 degree is 4o 3a + 7ab – 2a²b + 16 degree is 3

Ordering Polynomials

• Ascending Order – ordering polynomials based on their exponents least to greatest. You will only look at one variable’s exponents, usually ‘x’, unless told otherwise.

• Descending Order – ordering polynomials based on their exponents greatest to least.

Ascending Example

• Arrange polynomials in ascending order

7x³+ 2x²- 11Look at the exponents of the variable, choose the lowest. When there isn’t a variable with a term, it’s like there is an exponent of zero.7x³+ 2x²- 11x°-11 + 2x²+ 7x³

Descending Example

• Arrange polynomials in descending order

6x²+ 5 – 8x – 2x³

6x²+ 5x°- 8x – 2x³

-2x³+ 6x²- 8x + 5

Now you try…

Arrange in descending order 9 + 4x³- 3x – 10x²

Arrange in ascending order 10y³- 4y + 16

Now you try…

Arrange in descending order 9 + 4x³- 3x – 10x² 4x³- 10x²- 3x + 9

Arrange in ascending order 10y³- 4y + 16 16 – 4y + 10y³

Adding Polynomials

• Group like terms and combine (add) the coefficients.

(3x²- 4x + 8) + (2x – 7x²- 5)

Adding Polynomials

• Group like terms and combine (add) the coefficients.

(3x²- 4x + 8) + (2x – 7x²- 5) 3x²- 4x + 8 + 2x – 7x²- 5 Find like terms

-4x²- 2x + 3 Combine like terms (add)

Adding Polynomials

• Group like terms and combine (add) the coefficients.

(3x²- 4x + 8) + (2x – 7x²- 5) 3x²- 4x + 8 + 2x – 7x²- 5 Find like terms

-4x²-2x + 3 Combine like terms (add)

• Final answers must be in descending order

Now You Try…

(-2x³+ 3x²- 15x + 3) + (7x²+ 9x – 10)

(4 – 6x²+ 12x) + (8x²- 3x – 5)

Now You Try…

(-2x³+ 3x²- 15x + 3) + (7x²+ 9x – 10)

-2x³+ 10x²- 6x – 7

(4 – 6x²+ 12x) + (8x²- 3x – 5)

2x²+ 9x -1

Subtracting Polynomials

• Distribute the negative (minus) to the second set of parenthesis (include EVERYTHING in the parenthesis).

(3n²+ 13n³+ 5n) – (7n – 4n³)

Subtracting Polynomials

• Distribute the negative (minus) to the second set of parenthesis.

(3n²+ 13n³+ 5n) – (7n – 4n³)

• This will change the signs of everything in the second set of parenthesis.

Subtracting Polynomials

• Distribute the negative (minus) to the second set of parenthesis.

(3n²+ 13n³+ 5n) – (7n – 4n³)

3n²+ 13n³+ 5n – 7n + 4n³• Next combine like terms to simplify.

Subtracting Polynomials

(3n²+ 13n³+ 5n) – (7n – 4n³) 3n²+ 13n³+ 5n – 7n + 4n³

3n²+ 13n³+ 5n – 7n + 4n³ 17n³+ 3n²- 2n• The answer should be in descending

order

Now You Try…

(4x²+ 8x – 2) – (-5x – 2x²+ 7)

(10a³- 2a²+ 12a) – (7a²+ 3a – 10)

Now You Try…

(4x²+ 8x – 2) – (-5x – 2x²+ 7) 4x²+ 8x - 2 + 5x + 2x²- 7 6x²+ 13x – 9

(10a³- 2a²+ 12a) – (7a²+ 3a – 10) 10a³- 2a²+ 12a – 7a²- 3a + 10 10a³- 9a²+ 9a + 10

Review Problems Before the Quiz

(-2x²- 4x + 7) + (8x²+ 10x – 8)

(5x – 3) – (3x²+ 5x – 10)

Review Problems Before the Quiz

(-2x²- 4x + 7) + (8x²+ 10x – 8) -2x²- 4x + 7 + 8x²+ 10x – 8 6x²+ 6x - 1

(5x – 3) – (3x²+ 5x – 10) 5x – 3 – 3x²- 5x + 10 -3x²+ 7

Multiplying a Polynomial by a Monomial

• Use the distributive property.• Multiply the monomial by

EVERYTHING in the parenthesis.• Don’t forget your rules for

multiplying like bases and exponents! You will add the exponents.

• Combine like terms when necessary

Multiplying a Polynomial by a Monomial

-2x²(3x²- 7x + 10) -6x + 14x³- 20x²

• The answer should be in descending order.

4

Now You Try…

2x(-6x²+ 7x – 10)

(4x²- 8x + 3)(-5x)

Now You Try…

2x(-6x²+ 7x – 10)

-12x³+ 14x²- 20x

(4x²- 8x + 3)(-5x)

-20x³+ 40x² - 15x

Using Multiple Operations in Polynomials

• Always follow PEMDAS

7x(3x²- 5x + 7) + 4x²(3x – 1)

Using Multiple Operations in Polynomials

• Always follow PEMDAS• Multiply before adding/subtracting

oUse distributive property

7x(3x²- 5x + 7) + 4x²(3x – 1)

21x³- 35x²+ 49x + 12x³- 4x²

Using Multiple Operations in Polynomials

• Always follow PEMDAS• Multiply before adding/subtracting

o Use the distributive property

• Next, combine like terms

7x(3x²- 5x + 7) + 4x²(3x – 1) 21x³- 35x²+ 49x + 12x³- 4x² 33x³- 39x²+ 49x• Answers should be in descending order

Now You Try…

-2x(10x²- 7x + 4) + 3(-2x²+ 6x)

4x²(3x – 15) – 3x(11x³+ 5x²- 10)

Now You Try…

-2x(10x²- 7x + 4) + 3(-2x²+ 6x) -20x³+ 14x²- 8x – 6x²+ 18x

4x²(3x – 15) – 3x(11x³+ 5x²- 10) 12x³- 60x²- 33x – 15x³+ 30x

4

Now You Try…

-2x(10x²- 7x + 4) + 3(-2x²+ 6x) -20x³+ 14x²- 8x – 6x²+ 18x -20x³+ 8x²+ 10x

4x²(3x – 15) – 3x(11x³+ 5x²- 10) 12x³- 60x²- 33x – 15x³+ 30x -33x – 3x³- 60x²+ 30x4

4

Multiplying Polynomials Using FOIL

• When multiplying a binomial with a binomial you can use the FOIL method to simplify.

FirstOuterInner Last• This is a way to remember to multiply

each term of the expression.

Multiplying Polynomials Using FOIL

• Multiply the First terms in each binomial

(1x + 1)(1x + 2) 1x²• Use the rules for monomials when

multiplying!• Don’t forget to put a “1” in before

the variable, if there is not a coefficient.

Multiplying Polynomials Using FOIL

• Multiply the Outer terms in each binomial

(1x + 1)(1x + 2) 1x²+ 2x • Use the rules for monomials when

multiplying!

Multiplying Polynomials Using FOIL

• Multiply the Inner terms in each binomial

(1x + 1)(1x + 2) 1x²+ 2x + 1x

• Use the rules for monomials when multiplying!

Multiplying Polynomials Using FOIL

• Multiply the Last terms in each binomial

(1x + 1)(1x + 2) 1x²+ 2x + 1x + 2

• Use the rules for monomials when multiplying!

Multiplying Polynomials Using FOIL

• Now combine like terms.

(1x + 1)(1x + 2) 1x²+ 2x + 1x + 2

• Always put your answer in descending order!

Multiplying Polynomials Using FOIL

• Now combine like terms.

(1x + 1)(1x + 2) 1x²+ 2x + 1x + 2 1x²+ 3x + 2

• Always put your answer in descending order!

Now You Try…

(2x – 5)(x + 4)

(-4x – 8)(3x – 2)

Now You Try…

(2x – 5)(x + 4) 2x²+ 8x – 5x - 20 2x² + 3x - 20

(-4x – 8)(3x – 2) -12x²+ 8x – 24x + 16 -12x²– 16x + 16

Special Products

• Square of a sum or difference

(4y + 5)²

• The ENTIRE polynomial has to be squared. (4y + 5)(4y + 5)

• Then use FOIL to solve.

Special Products

• Square of a sum or difference

(4y + 5)² (4y + 5)(4y + 5) 16y²+ 20y + 20y + 25

• Combine like terms

Special Products

• Square of a sum or difference

(4y + 5)² (4y + 5)(4y + 5) 16y²+ 20y + 20y + 25 16y²+ 40y + 25• Final answer should be in descending

order.

Special Products

• Product of a sum and a difference• The binomials are the same except

one is plus and one is minus. (3n + 2)(3n – 2)

• Use FOIL to simplify

Special Products

• Product of a sum and a difference (3n + 2)(3n – 2) 9n²- 6n + 6n - 4 9n²- 4• Combine like terms• Notice that the two center terms (like

terms) will cancel each other out.

Now You Try…

(8c + 3d)²

(5m³- 2n)²

(11v – 8w)(11v + 8w)

Now You Try…

(8c + 3d)² (8c + 3d)(8c + 3d) 64c²+ 24cd + 24cd + 9d² 64c²+ 48cd + 9d²

(5m³- 2n)² (5m³- 2n)(5m³- 2n) 25m – 10m³n – 10m³n + 4n² 25m – 20m³n + 4n²

6

6

Now You Try…

(11v – 8w)(11v + 8w) 11v²+ 88vw – 88vw – 64w² 121v²- 64w²

Multiplying Polynomials

• Use the distributive property when multiplying polynomials. Multiply everything in the first set of parenthesis by everything in the second set of parenthesis.

(4x + 9)(2x²- 5x + 3)

Multiplying Polynomials

• Multiply the first term by everything in the second set of parenthesis.

(4x + 9)(2x²- 5x + 3) 8x³- 20x²+ 12x

Multiplying Polynomials

• Multiply the second term by everything in the second set of parenthesis.

(4x + 9)(2x²- 5x + 3) 8x³- 20x²+ 12x + 18x²- 45x + 27

Multiplying Polynomials

• Combine like terms. (4x + 9)(2x²- 5x + 3) 8x³- 20x²+ 12x + 18x²- 45x + 27 8x³- 2x²- 33x + 27

• Final answer should be in descending order.

Now You Try…

(y²- 2y + 5)(6y²- 3y + 1)

Now You Try…

(y²- 2y + 5)(6y²- 3y + 1)

6y – 3y³+ y²- 12y³+ 6y²- 2y + 30y²- 15y + 5 6y – 15y³+ 37y²- 17y + 5

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Dividing Polynomials

• Dividing polynomials is the same as dividing monomials except there is more than one term.

• Subtract the exponents of like bases and simplify the coefficients by dividing.

• Do not give the answer in a decimal.• Cannot have negative exponents!• Answers should be in descending order.

Dividing Polynomials

6x³- 4x²+ 2x 2x

• The problem could be looked at like three separate problems or as one. Just take each term separately.

Dividing Polynomials

6x³- 4x²+ 2x 2x

6x³ -4x² 2x 2x 2x 2x • Simplify each

Dividing Polynomials

6x³- 4x²+ 2x 2x

6x³ -4x² 2x 2x 2x 2x

3x²- 2x + 1• The signs stay the same as the original

problem, unless they change when simplified.

Your turn…

10x³+ 15x²- 25x 5x

-36x³- 24x²+ 12x -6x

Your turn…

10x³+ 15x²- 25x 5x 2x²+ 3x - 5

-36x³- 24x²+ 12x -6x 6x²+ 4x - 2