Polynomials Terms and Factoring Algebra I H.S. Created by: Buddy L. Anderson Algebra I H.S. Created by: Buddy L. Anderson

Vocabulary / Monomial: A number, a variable or the product of a number and one or more variables / Polynomial: A monomial or a sum of monomials. / Binomial: A polynomial with exactly two terms. / Trinomial: A polynomial with exactly three terms. / Coefficient: A numerical factor in a term of an algebraic expression. / Degree of a monomial: The sum of the exponents of all of the variables in the monomial. / Monomial: A number, a variable or the product of a number and one or more variables / Polynomial: A monomial or a sum of monomials. / Binomial: A polynomial with exactly two terms. / Trinomial: A polynomial with exactly three terms. / Coefficient: A numerical factor in a term of an algebraic expression. / Degree of a monomial: The sum of the exponents of all of the variables in the monomial.

Vocabulary / Degree of a polynomial in one variable: The largest exponent of that variable. / Standard form: When the terms of a polynomial are arranged from the largest exponent to the smallest exponent in decreasing order. / Degree of a polynomial in one variable: The largest exponent of that variable. / Standard form: When the terms of a polynomial are arranged from the largest exponent to the smallest exponent in decreasing order.

Degree of a Monomial / What is the degree of the monomial? / The degree of a monomial is the sum of the exponents of the variables in the monomial. / The exponents of each variable are 4 and =6. / Therefore the degree is six and it can be referred to as a sixth degree monomial. / What is the degree of the monomial? / The degree of a monomial is the sum of the exponents of the variables in the monomial. / The exponents of each variable are 4 and =6. / Therefore the degree is six and it can be referred to as a sixth degree monomial.

Polynomial / A polynomial is a monomial or the sum of monomials / Each monomial in a polynomial is a term of the polynomial. / The number factor of a term is called the coefficient. The coefficient of the first term in a polynomial is the lead coefficient / A polynomial with two terms is called a binomial. / A polynomial with three terms is called a trinomial. / A polynomial is a monomial or the sum of monomials / Each monomial in a polynomial is a term of the polynomial. / The number factor of a term is called the coefficient. The coefficient of the first term in a polynomial is the lead coefficient / A polynomial with two terms is called a binomial. / A polynomial with three terms is called a trinomial.

Degree of a Polynomial in One Variable / The degree of a polynomial in one variable is the largest exponent of that variable. / The degree of this polynomial is 2, since the highest exponent of the variable x is 2. /T/The degree of a polynomial in one variable is the largest exponent of that variable. /T/The degree of this polynomial is 2, since the highest exponent of the variable x is 2.

Standard Form of a Polynomial / To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term. / The leading coefficient, the coefficient of the first term in a polynomial written in standard form, should be positive. / To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term. / The leading coefficient, the coefficient of the first term in a polynomial written in standard form, should be positive.

Put in Standard Form

Factoring Polynomials / By Grouping / Difference of Squares / Perfect Square Trinomials / X-Box Method / By Grouping / Difference of Squares / Perfect Square Trinomials / X-Box Method

By Grouping / When polynomials contain four terms, it is sometimes easier to group like terms in order to factor. / Your goal is to create a common factor. / You can also move terms around in the polynomial to create a common factor. / When polynomials contain four terms, it is sometimes easier to group like terms in order to factor. / Your goal is to create a common factor. / You can also move terms around in the polynomial to create a common factor.

By Grouping  FACTOR: 3xy - 21y + 5x – 35 / Factor the first two terms: 3xy - 21y = 3y (x – 7) / Factor the last two terms: + 5x - 35 = 5 (x – 7)  The terms in the parentheses are the same so it ’ s the common factor Now you have a common factor (x - 7) (3y + 5)  FACTOR: 3xy - 21y + 5x – 35 / Factor the first two terms: 3xy - 21y = 3y (x – 7) / Factor the last two terms: + 5x - 35 = 5 (x – 7)  The terms in the parentheses are the same so it ’ s the common factor Now you have a common factor (x - 7) (3y + 5)

By Grouping  FACTOR: 15x – 3xy + 4y – 20 / Factor the first two terms: 15x – 3xy = 3x (5 – y) / Factor the last two terms: + 4y – 20 = 4 (y – 5) / The terms in the parentheses are opposites so change the sign on the (-y + 5) or – 4 (5 - y)  Now you have a common factor (5 – y) (3x – 4)  FACTOR: 15x – 3xy + 4y – 20 / Factor the first two terms: 15x – 3xy = 3x (5 – y) / Factor the last two terms: + 4y – 20 = 4 (y – 5) / The terms in the parentheses are opposites so change the sign on the (-y + 5) or – 4 (5 - y)  Now you have a common factor (5 – y) (3x – 4)

Difference of Squares / When factoring using a difference of squares, look for the following three things: / only 2 terms / minus sign between them / both terms must be perfect squares / If all 3 of the above are true, write two ( ), one with a + sign and one with a – sign : ( + ) ( - ). / When factoring using a difference of squares, look for the following three things: / only 2 terms / minus sign between them / both terms must be perfect squares / If all 3 of the above are true, write two ( ), one with a + sign and one with a – sign : ( + ) ( - ).

Try These  1. a 2 – 16  2. x 2 – 25  3. 4y 2 – 16  4. 9y 2 – 25  5. 3r 2 – 81 / 6. 2a  1. a 2 – 16  2. x 2 – 25  3. 4y 2 – 16  4. 9y 2 – 25  5. 3r 2 – 81 / 6. 2a

Perfect Square Trinomials / When factoring using perfect square trinomials, look for the following three things: / 3 terms / last term must be positive / first and last terms must be perfect squares / If all three of the above are true, write one ( ) 2 using the sign of the middle term. / When factoring using perfect square trinomials, look for the following three things: / 3 terms / last term must be positive / first and last terms must be perfect squares / If all three of the above are true, write one ( ) 2 using the sign of the middle term.

Try These  1. a 2 – 8a + 16 / 2. x x + 25 / 3. 4y y + 16 / 4. 9y y + 25  5. 3r 2 – 18r + 27 / 6. 2a 2 + 8a - 8  1. a 2 – 8a + 16 / 2. x x + 25 / 3. 4y y + 16 / 4. 9y y + 25  5. 3r 2 – 18r + 27 / 6. 2a 2 + 8a - 8

X-Box Method / No, we are not going to feed polynomials into a game system that will factor them. / We will go over the following example. / No, we are not going to feed polynomials into a game system that will factor them. / We will go over the following example.

X-Box Method -13 (3)(-10)= x 2x 3x 2 x-5 3x +2 3x 2 -13x -10 = (x-5)(3x+2)

X-Box Method / The color codes in the equation show where the numbers go in the diamond and box. / The -15 and 2 came from the fact that you needed 2 numbers that multiplied to get -30 and added to get -13. / The color codes in the equation show where the numbers go in the diamond and box. / The -15 and 2 came from the fact that you needed 2 numbers that multiplied to get -30 and added to get -13.

X-Box Method / The outside of the box are the GCF of what they are above or beside. / These give you you r 2 factors. / The outside of the box are the GCF of what they are above or beside. / These give you you r 2 factors.