PRC Ch P_3.notebook How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials How to remove common factors from polynomials How to factor special polynomial forms How to factor trinomials as the product of two binomials How to factor by grouping Aug 1 2:17 PM Aug 1 2:24 PM 1
The definition of a polynomial How to identify the various characteristics of a polynomial How to write a polynomial in standard form How to identify like terms How to add and subtract polynomials How to muliply two polynomials Aug 1 2:17 PM The most common type of algebraic expression Examples: (polynomial in x) (polynomial in x) (polynomial in x and y) terms: coefficient: " " have the form degree: of a term " " of the polynomial " " Aug 1 2:40 PM 2
Definition of a Polynomial in x Let a 0, a 1, a 2,..., a n be real numbers where n is a non negative integer. Then a polynomial in x is an expression of the form a n x n + a n 1 x n 1 +... + a 1 x + a 0 where a n = 0 degree of polynomial: n leading coefficient: a n constant term: a 0 monomials: binomials: trinomials: Aug 1 3:10 PM standard form of a polynomial: a polynomial is in standard form when the terms are in order from highest degree to lowest degree (left to right) Aug 1 3:25 PM 3
Example Determine if the following are polynomials. a. b. c. NOTE: The zero polynomial: all coefficients are zero denoted as simply "0" considered to have 0 degree Aug 1 3:49 PM Like terms of polynomials can be combined through addition and subtraction. like terms: terms having the same variables to the same powers combine by the distributive property Aug 1 4:36 PM 4
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Finding the product of two polynomials Finding the product of two polynomials is done using the Distributive Property. Aug 1 4:58 PM Aug 1 5:14 PM 6
Example Multiplying Polynomials: Two Trinomials Find the product of the two trinomials. Aug 1 3:49 PM Aug 1 5:28 PM 7
SUMMARY: The definition of a polynomial terms variable constant like terms monomials, binomials, trinomials coefficients (real numbers) leading coefficient exponents (non negative integers) degree of a term degree of the polynomial standard form Aug 1 5:16 PM SUMMARY: Arithmetic Operations of Polynomials Addition and Subtraction Identify like terms Combine like terms using the Distributive Property Multiplication General Method Use Distributive Property Multiply every term in one polynomial by every term in the other. Result of General Method for Binomials F.O.I.L. Aug 1 5:16 PM 8
How to recognize and use the Special Product forms: Binomial Conjugates Binomial Squared Binomial Cubed Aug 1 2:17 PM Special Products are polynomial products that: occur frequently in algebra and... follow an easily recognizable pattern that... allows the result to be written down without performing the calculation For the following Special Products let: u and v be real numbers, variables, or algebraic expressions Aug 2 11:38 AM 9
The Product of Conjugate Binomials Aug 2 12:15 PM The Square of a Binomial Aug 2 12:15 PM 10
The Cube of a Binomial Aug 2 12:15 PM Aug 2 12:35 PM 11
Aug 2 4:49 PM SUMMARY: Special Products of Polynomials Commonly occur in algebra The result of the product follows a pattern. This pattern allows the result of the product to be written down without having to perform the actual operations. Aug 1 5:16 PM 12
SUMMARY: Some Special Products of Polynomials The product of conjugate binomials The square of a binomial The cube of a binomial Aug 1 5:16 PM The general formula for expanding a binomial to some power The general pattern Finding coefficients using Pascal's triangle Finding coefficients using the formula Aug 1 2:17 PM 13
General Formula for Binomial Expansion Pattern for expanding a binomial to some power: Aug 2 4:55 PM General Formula for Binomial Expansion Pattern for expanding a binomial to some power: where C r are the coefficients from Pascal's Triangle Aug 2 4:55 PM 14
Binomial Expansion Coefficients From Pascal's Triangle The 1st row (top) has one number. The 2nd row has two numbers. The 3rd row has three numbers....and so on... The centers of every row are vertically aligned so the numbers in each row are below and between the numbers in the row above it. Each row begins with a "1" and ends with a "1". The "inner" numbers of each new row are found by adding the numbers on either side of it in the row above. Aug 2 4:58 PM Binomial Expansion Coefficients From Formula C r = n C r where n C r = Let's try a few: and r = 0, 1, 2, 3,..., n Aug 2 4:58 PM 15
Example Using the Formula to Find Pascal's Coefficients Use the formula to find Pascal's coefficients for the following expansion. Verify these results using Pascal's triangle. Aug 2 6:25 PM Example Expanding Binomials Raised to a Power Use the coefficients found in the previous example to complete the binomial expansion. Aug 2 6:25 PM 16
Example Expanding Binomials Raised to a Power Use the previous example to write the same expansion for the binomial of the difference of the same two terms. Aug 2 6:25 PM SUMMARY: General Form for Expanding a Binomial to Some Power Coefficents of Expansion The coefficients are determined by the degree of the expansion. A pattern for these coefficients was identified by Blaise Pascal Pascal's triangle. Formula for coefficients of expansion: C r = n C r Aug 1 5:16 PM 17
Why factoring polynomials is important How to identify when a polynomial is fully factored How to remove common factors Aug 1 2:17 PM Factoring Multiplying product Factoring factors For Numbers: 6 = (2)(3) For Polynomials: Aug 3 10:55 AM 18
Why is factoring polynomials important? The Zero Factor Property: Aug 3 3:44 PM When is Factoring Complete? For Numbers: 24=(1)(24) 24=(2)(12) 24=(3)(8) 24=(4)(6) Aug 3 10:55 AM 19
When is Factoring Complete? For Polynomials: Aug 3 2:45 PM Aug 3 2:57 PM 20
Factoring Polynomials Removing Common Factors (using the Distributive Property) Aug 3 3:53 PM SUMMARY: General Factoring Principles Factoring is the opposite of multiplying Prime Factoring Prime means irreducible ie. no factors can be factored any further Factoring Polynomials Polynomials include a variable Many equations are in the form of polynomials Factoring is the method for solving polynomial equations (via the Zero Factor property). Prime (or Irreducible) over the: Integers (default interpretation) Rationals Reals Aug 1 5:16 PM 21
How to recognize three special polynomial factoring forms How to use these three special polynomial factoring forms Aug 1 2:17 PM Special Polynomial Factoring Forms The Difference of Two Squares Perfect Square Trinomials The Sum or Difference of Two Cubes Aug 3 4:14 PM 22
The Difference of Two Squares Example: Aug 4 12:31 PM Perfect Square Trinomials Examples: Aug 4 12:35 PM 23
The Sum or Difference of Two Cubes Examples: Aug 4 12:35 PM Aug 4 12:40 PM 24
a. b. Aug 4 1:05 PM a. b. Aug 4 1:09 PM 25
Aug 4 3:14 PM SUMMARY: Special Factoring Forms for Polynomials Remove any common factors first. Difference of Two Squares Perfect Square Trinomials Sum or Difference of Two Cubes Aug 1 5:16 PM 26
How to factor trinomials that have binomial factors Aug 1 2:17 PM Trinomials with Binomial Factors Option 1 Consider a trinomial of the form that is the product of two binomials. For simplicity, let a, b, c, d, e, f, and g all be positive integers. Aug 4 3:40 PM 27
Trinomials with Binomial Factors Option 2 Consider a trinomial of the form that is the product of two binomials. For simplicity, let a, b, c, d, e, f, and g all be positive integers. Aug 4 3:40 PM Trinomials with Binomial Factors Option 3 Consider a trinomial of the form that is the product of two binomials. For simplicity, let a, b, c, d, e, f, and g all be positive integers. Aug 4 3:40 PM 28
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SUMMARY: Factoring Trinomials with Binomial Factors Aug 1 5:16 PM How to factor a four term polynomial by grouping How to factor a trinomial by grouping Aug 1 2:17 PM 30
Factoring by Grouping In some cases, a four term polynomial can be factored by grouping the four terms into two groups of two factors. The necessary elements are that: one or both of the two terms have a common factor the quantity left behind after removing the common factor(s) is the same Aug 4 5:15 PM...lets see how this works. Aug 4 5:15 PM 31
Can factoring by grouping be used to factor a trinomial? factors of a factors of c "b" is the additive combination of factors of "a" and "c" "b" is the subtractive combination of factors of "a" and "c" Aug 4 5:43 PM Aug 5 10:36 AM 32
SUMMARY: Factoring by Grouping Four Term Polynomials Trinomials CAUTION: Be careful when using the Associative Property to create a grouping with a negative leading term Aug 1 5:16 PM 33