STANDARD 3 & 4: POLYNOMIAL OPERATIONS & FACTORING
Polynomial Vocabulary
Polynomial- One or more terms put together by addition and subtraction
Monomials – one term (example 4x)
Binomials- two terms (example 5x – 2)
Trinomials- three terms (example: 3x2 – 4x + 8)
Monomials – one term (example 4x)
Binomials- two terms (example 5x – 2)
Trinomials- three terms (example: 3x2 – 4x + 8)
Adding & Subtracting Polynomials
Adding Polynomials
1. Combine like terms
**tip: Remember that x and x2 are not like-terms.
Example: Add (-5x + 6) and (4 – 3x)
Combine like terms -5x + – 3x and 6 + 4
-8x + 10
How to Subtract Polynomials
1. Distribute the negative 1
2. Combine like terms
Example: (9x – 4) – (-3x + 3)
9x – 4 + 3x - 3 distribute the negative
12x – 7 combine like-terms
1. Combine like terms
**tip: Remember that x and x2 are not like-terms.
Example: Add (-5x + 6) and (4 – 3x)
Combine like terms -5x + – 3x and 6 + 4
-8x + 10
How to Subtract Polynomials
1. Distribute the negative 1
2. Combine like terms
Example: (9x – 4) – (-3x + 3)
9x – 4 + 3x - 3 distribute the negative
12x – 7 combine like-terms
Multiplying Polynomials
Multiplying polynomials
1. Use the distributive property
2. Combine like terms
Example: (9x – 4)(-3x + 3) ** since there is no sign in between the binomials, we multiply!
-27x2 + 27 – 12x – 12 distributive property
-27x2 – 12x + 15 combine like terms
1. Use the distributive property
2. Combine like terms
Example: (9x – 4)(-3x + 3) ** since there is no sign in between the binomials, we multiply!
-27x2 + 27 – 12x – 12 distributive property
-27x2 – 12x + 15 combine like terms
Exponent Properties are Important to Use!
All Operations Combined
** Always look what operations the problem shows.
**One common mistake that a lot of students make is distributing/multiplying when there is a subtraction sign. Do not make this mistake!
**Always follow the directions and the Order of Operations
PEMDAS!! (Parenthesis, Exponents, Multiply, Divide, Add Subtract)
**One common mistake that a lot of students make is distributing/multiplying when there is a subtraction sign. Do not make this mistake!
**Always follow the directions and the Order of Operations
PEMDAS!! (Parenthesis, Exponents, Multiply, Divide, Add Subtract)
Word Problems involving Polynomials
Example: The side of a cube is represented by x + 1. Find, in terms of x, the volume of the cube.
Volume = length•width•height
Volume = (x+1)(x+1)(x+1) distribute
= (x+1)(x²+2x+1) distribute
= x³+3x²+3x+1 cubic units
Volume = length•width•height
Volume = (x+1)(x+1)(x+1) distribute
= (x+1)(x²+2x+1) distribute
= x³+3x²+3x+1 cubic units
Long Division of Polynomials
How to do Long division
***it's just like regular long division:
1. Divide the first term by the first term (write it on top)
2. Multiply the quotient by the whole divisor (write it underneath)
3. Subtract
4. Bring down the next term
5. Repeat the process
***it's just like regular long division:
1. Divide the first term by the first term (write it on top)
2. Multiply the quotient by the whole divisor (write it underneath)
3. Subtract
4. Bring down the next term
5. Repeat the process
Check out this video example:
http://virtualnerd.com/algebra-1/rational-expressions-functions/polynomial-long-division-example.php
Factoring!
Teach me how to Factor: http://www.schooltube.com/video/5a35f2564c0af3946c3e/
Factoring Trinomial Quadratics
For simple factoring in class, we used the x-method. You can use this method or any other that you know and feel comfortable with.
Example: Factor x2 – 7x + 6. (that is x-squared)
In this case, I am multiplying to a positive six, so the factors are either both positive or both negative. I am adding to a negative seven, so the factors are both negative. The factors of 6 that add up to 7 are 1 and 6, so I will use –1 and –6:
x2 – 7x + 6 = (x – 1)(x – 6)
Example: Factor x2 + x – 6. (again, that is x-squared)
Since I am multiplying to a negative six, I need factors of opposite signs; that is, one factor will be positive and the other will be negative. The larger factor (in absolute value) will get the "plus" sign, because I am adding to a positive 1. Since these opposite-signed numbers will be adding to 1, I need the two factors to be one unit apart. The factor pairs for six are 1 and 6, and 2 and 3. The second pair are one apart, so I want to use 2 and 3, with the 3 getting the "plus" sign (so the 2 gets the "minus" sign).
x2 + x – 6 = (x – 2)(x + 3). (x-squared)
Watch the video on Factoring a trinomial when a = 1 http://www.virtualnerd.com/algebra-1/polynomials-and-factoring/trinomial-factorization-example.php
When the a value does not equal one- you can still use the x-method, but remember there is one extra step! Divide each side of the X by the a-value and simplify, to create the factors!
Example: Factor x2 – 7x + 6. (that is x-squared)
In this case, I am multiplying to a positive six, so the factors are either both positive or both negative. I am adding to a negative seven, so the factors are both negative. The factors of 6 that add up to 7 are 1 and 6, so I will use –1 and –6:
x2 – 7x + 6 = (x – 1)(x – 6)
Example: Factor x2 + x – 6. (again, that is x-squared)
Since I am multiplying to a negative six, I need factors of opposite signs; that is, one factor will be positive and the other will be negative. The larger factor (in absolute value) will get the "plus" sign, because I am adding to a positive 1. Since these opposite-signed numbers will be adding to 1, I need the two factors to be one unit apart. The factor pairs for six are 1 and 6, and 2 and 3. The second pair are one apart, so I want to use 2 and 3, with the 3 getting the "plus" sign (so the 2 gets the "minus" sign).
x2 + x – 6 = (x – 2)(x + 3). (x-squared)
Watch the video on Factoring a trinomial when a = 1 http://www.virtualnerd.com/algebra-1/polynomials-and-factoring/trinomial-factorization-example.php
When the a value does not equal one- you can still use the x-method, but remember there is one extra step! Divide each side of the X by the a-value and simplify, to create the factors!
Special Patterns
How to recognize a difference of squares:: http://www.virtualnerd.com/algebra-1/polynomials-and-factoring/difference-of-squares-identification.php
Factoring using the difference of squares pattern: http://www.virtualnerd.com/algebra-1/polynomials-and-factoring/factoring-special-products/difference-square-or-cubes/difference-of-squares-example
Sum & Difference of Cubes
Standard 3 Practice | |
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Standard 4 Practice | |
File Size: | 66 kb |
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