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Polynomials Algebra I

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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.

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**Examples Monomial Binomial Trinomial 13n -5z 2a + 3c 6x²+ 3xy**

p²+ 5p + 4 3a²- 2ab - b²

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**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.

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**Examples Monomial Polynomial 8y³ degree is 3 3a degree is 1**

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

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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.

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**Ascending Example Arrange polynomials in ascending order 7x³+ 2x²- 11**

Look 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° x²+ 7x³

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**Descending Example Arrange polynomials in descending order**

6x²+ 5 – 8x – 2x³ 6x²+ 5x°- 8x – 2x³ -2x³+ 6x²- 8x + 5

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Now you try… Arrange in descending order 9 + 4x³- 3x – 10x² Arrange in ascending order 10y³- 4y + 16

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Now you try… Arrange in descending order 9 + 4x³- 3x – 10x² 4x³- 10x²- 3x + 9 Arrange in ascending order 10y³- 4y – 4y + 10y³

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Adding Polynomials Group like terms and combine (add) the coefficients. (3x²- 4x + 8) + (2x – 7x²- 5)

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Adding Polynomials Group like terms and combine (add) the coefficients. (3x²- 4x + 8) + (2x – 7x²- 5) 3x²- 4x x – 7x²- 5 Find like terms -4x²- 2x Combine like terms (add)

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Adding Polynomials Group like terms and combine (add) the coefficients. (3x²- 4x + 8) + (2x – 7x²- 5) 3x²- 4x x – 7x²- 5 Find like terms -4x²-2x Combine like terms (add) Final answers must be in descending order

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Now You Try… (-2x³+ 3x²- 15x + 3) + (7x²+ 9x – 10) (4 – 6x²+ 12x) + (8x²- 3x – 5)

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Now You Try… (-2x³+ 3x²- 15x + 3) + (7x²+ 9x – 10) -2x³+ 10x²- 6x – 7 (4 – 6x²+ 12x) + (8x²- 3x – 5) 2x²+ 9x -1

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**Subtracting Polynomials**

Distribute the negative (minus) to the second set of parenthesis (include EVERYTHING in the parenthesis). (3n²+ 13n³+ 5n) – (7n – 4n³)

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**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.

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**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.

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**Subtracting Polynomials**

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

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Now You Try… (4x²+ 8x – 2) – (-5x – 2x²+ 7) (10a³- 2a²+ 12a) – (7a²+ 3a – 10)

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Now You Try… (4x²+ 8x – 2) – (-5x – 2x²+ 7) 4x²+ 8x x + 2x²- 7 6x²+ 13x – 9 (10a³- 2a²+ 12a) – (7a²+ 3a – 10) 10a³- 2a²+ 12a – 7a²- 3a a³- 9a²+ 9a + 10

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**Review Problems Before the Quiz**

(-2x²- 4x + 7) + (8x²+ 10x – 8) (5x – 3) – (3x²+ 5x – 10)

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**Review Problems Before the Quiz**

(-2x²- 4x + 7) + (8x²+ 10x – 8) -2x²- 4x x²+ 10x – 8 6x²+ 6x - 1 (5x – 3) – (3x²+ 5x – 10) 5x – 3 – 3x²- 5x x²+ 7

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**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

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**Multiplying a Polynomial by a Monomial**

-2x²(3x²- 7x + 10) -6x + 14x³- 20x² The answer should be in descending order.

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Now You Try… 2x(-6x²+ 7x – 10) (4x²- 8x + 3)(-5x)

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Now You Try… 2x(-6x²+ 7x – 10) -12x³+ 14x²- 20x (4x²- 8x + 3)(-5x) -20x³+ 40x² - 15x

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**Using Multiple Operations in Polynomials**

Always follow PEMDAS 7x(3x²- 5x + 7) + 4x²(3x – 1)

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**Using Multiple Operations in Polynomials**

Always follow PEMDAS Multiply before adding/subtracting Use distributive property 7x(3x²- 5x + 7) + 4x²(3x – 1) 21x³- 35x²+ 49x + 12x³- 4x²

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**Using Multiple Operations in Polynomials**

Always follow PEMDAS Multiply before adding/subtracting 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

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Now You Try… -2x(10x²- 7x + 4) + 3(-2x²+ 6x) 4x²(3x – 15) – 3x(11x³+ 5x²- 10)

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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

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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+ 30x

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**Multiplying Polynomials Using FOIL**

When multiplying a binomial with a binomial you can use the FOIL method to simplify. First Outer Inner Last This is a way to remember to multiply each term of the expression.

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**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.

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**Multiplying Polynomials Using FOIL**

Multiply the Outer terms in each binomial (1x + 1)(1x + 2) 1x²+ 2x Use the rules for monomials when multiplying!

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**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!

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**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!

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**Multiplying Polynomials Using FOIL**

Now combine like terms. (1x + 1)(1x + 2) 1x²+ 2x + 1x + 2 Always put your answer in descending order!

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**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!

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Now You Try… (2x – 5)(x + 4) (-4x – 8)(3x – 2)

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Now You Try… (2x – 5)(x + 4) 2x²+ 8x – 5x x² + 3x - 20 (-4x – 8)(3x – 2) -12x²+ 8x – 24x x²– 16x + 16

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**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.

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**Special Products Square of a sum or difference (4y + 5)²**

(4y + 5)(4y + 5) 16y²+ 20y + 20y + 25 Combine like terms

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**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.

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**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

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**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.

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Now You Try… (8c + 3d)² (5m³- 2n)² (11v – 8w)(11v + 8w)

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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

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Now You Try… (11v – 8w)(11v + 8w) 11v²+ 88vw – 88vw – 64w² 121v²- 64w²

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**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)

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**Multiplying Polynomials**

Multiply the first term by everything in the second set of parenthesis. (4x + 9)(2x²- 5x + 3) 8x³- 20x²+ 12x

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**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

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**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.

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Now You Try… (y²- 2y + 5)(6y²- 3y + 1)

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**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.

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**Dividing Polynomials 6x³- 4x²+ 2x 2x**

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

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**Dividing Polynomials 6x³- 4x²+ 2x 2x 6x³ -4x² 2x 2x 2x 2x**

Simplify each

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**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.

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Your turn… 10x³+ 15x²- 25x 5x -36x³- 24x²+ 12x -6x

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Your turn… 10x³+ 15x²- 25x 5x 2x²+ 3x x³- 24x²+ 12x -6x 6x²+ 4x - 2

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