Polynomials and Factoring Chapter 8 Polynomials and Factoring
Definitions Binomial- a polynomial of two terms Degree of polynomial- the greatest value of exponent of any term of the polynomial Degree of monomial- the sum of the exponents of the variables of a monomial Difference of two squares- a difference of two squares is an expression of the form a²-b². Factors to (a + b)(a - b)
Definitions Factoring by grouping- a method of factoring that uses the distributive property to remove a common binomial factor of two pairs of terms Monomial- a real number, a variable, or a product of a real number and one or more variables with whole-number exponents
Definitons Perfect square trinomial- any trinomial of the form a² + 2ab + b² or a² - 2ab + b² Polynomial- a monomial of the sum or difference of two or more monomials. A quotient with a variable in the denominator is not a polynomial Standard form of a polynomial- the form of a polynomial that places the terms in descending order by degree Trinomial- a polynomial of three terms
Adding and subtracting polynomials 8.1 Adding and subtracting polynomials
Essential Understanding You can use monomials to form larger expressions called polynomials. Polynomials can be added or subtracted.
Degree of monomial The degree of a monomial is the sum of the exponents of its variables only. The degree of a nonzero constant is 0. Zero means it has no degree
Finding the Degree of Monomial Examples: 11 ⅞ 12 ½ ∏ 92 Any polynomial whose largest term has an exponent that is not stated, and no variables has a degree of zero
Finding the Degree of Monomial Examples: x z a y b p Any polynomial whose largest term has a variable with an implicit exponent of one has a degree of one
Finding the Degree of Monomial Examples: Degree of 2 2xy 4z² 7pq Degree of 3 xyz x³ Xy² Degree of 4 x²y² wxyz Any mononomial whose largest term has an exponent or has multiple terms with exponents has the sum of the exponents as its degree
Combining Like Monomials Example Explanation To find degree, sum exponents Even if bases are not the same x²x² 2+2=4 6x³y² 3+2=5 -7x⁴z² 4+2=6 uvwxyz 1+1+1+1+1+1=6 Combine exponents Term has a degree of four Term has a degree of five Term has a degree of six
Standard Form of Polynomial Examples in Standard Form Examples NOT in Standard Form In standard form, the exponents (degree) descend as the terms are listed. 5x² + 7x - 10 12x³ - 6x + 9 19x⁴ - 8x³ - 5x x⁴ + x³ + x² + x + 1 An expression is not in standard form if the degrees do not descend in order 6x-12x² 1-3x⁴+2x+x³ 12 + 12x² + 12x⁴ + 12x
Degree of a Polynomial The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent. The degree of 3x⁴ + 5x² - 7x + 1 is four
Degree of Polynomial Polynomial Degree Name of Degree Number of Terms Name using Number of Terms 10 None (0) Constant One monomial 9237238 x One (1) Linear 7x-5 Two binomial 5x² Two (2) Quadratic 12x²+144 3x²+9x+12 Three trinomial 8x³ Three (3) Cubic 4x³+10x x³+3x²+x+1 Four polynomial x⁴+x³+x²+x+1 Four (4) Quartic Five
Classifying Polynomials To classify a polynomial Ensure all like terms are combined Find the term with the greatest exponent List term with greatest exponent Find term with next greatest exponent List term with next greatest exponent Continue process until you reach the constant term or, if no constant term exists, the term with the least greatest exponent
Multiplying and factoring 8.2 Multiplying and factoring
Essential Understanding You can use the distributive property to multiply a monomial by a polynomial
Multiplying Polynomials
Greatest Common Factor
Multiplying binomials 8.3 Multiplying binomials
Essential Understanding There are several ways to find the product of two binomials including models, algebra, and tables.
Multiplying Binomials
Multiplying Binomials Using the Distributive Property
Multiplying Binomials Using a Table
FOIL FOIL stands for first outer inner last This method does not work for multiplying two polynomials with more than two terms for each
Multiplying special cases 8.4 Multiplying special cases
Essential Understanding There are special rules you can use to simplify the square of a binomial or the product of a sum and difference. Squares of binomials have two forms: (a + b)² (a – b)²
The Square of a Binomial The square of a binomial is the square o the first term plus twice the product of the two terms plus the square of the last term. (a + b)² = a² + 2ab + b² (a – b)² = a² -2ab + b²
The Product of a sum and difference The product of the sum and difference of the same two terms is the difference o their squares. (a + b)(a – b) = a² - b²
8.5 Factoring x²+bx+c
Essential Understanding You can write some trinomials of the form x² + bx + c as the product of two binomials
Factoring x² + bx + c Use a table to list the pairs of factors of the constant term c and the sums of those pairs of factors.
8.6 Factoring ax²+bx+c
Essential Understanding You can write some trinomials of the form ax² + bx + c as the product of two binomials
Factoring When a∙c is Positive
Factoring When a∙c is Negative
Factoring Special Cases 8.7 Factoring Special Cases
Essential Understanding You can factor some trinomials by “reversing” the rules for multiplying special case binomials that you learned in lesson 8.4
Factoring Perfect Square Trinomials Any trinomial of the form a² + 2ab + b² is a perfect square trinomial because it is the result of squaring a binomial. For example let a,b be real numbers: (a + b)² = (a + b)(a + b) = a² + 2ab + b² (a - b)² = (a - b)(a - b) = a² - 2ab + b²
Factoring a Difference of Two Squares For all real numbers a,b: a² - b² = (a + b)(a – b)
8.8 Factoring by grouping
Essential Understanding Some polynomials of a degree greater than 2 can be factored
Factoring Polynomials Factor out the greatest common factor (GCF) If the polynomial has two terms or three terms, look for a difference of two squares, a perfect square trinomial or a pair of binomial factors If the polynomial has four or more terms, group terms and factor to find common binomial factors. As a final check, make sure there are no common factors other than 1