polynomials algebra i. vocabulary monomial – a number, variable or a product of a number and one...
TRANSCRIPT
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
4
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