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Polynomials Terms and Factoring

Polynomials Terms and Factoring. 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.

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Polynomials Terms and Factoring

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

  2. 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.

  3. 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.

  4. 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 2. 4+2=6. • Therefore the degree is six and it can be referred to as a sixth degree monomial.

  5. 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.

  6. 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.

  7. 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.

  8. Put in Standard Form

  9. Factoring Polynomials • By Grouping • Difference of Squares • Perfect Square Trinomials • X-Box Method

  10. 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.

  11. 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)

  12. 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 4 - 4 (-y + 5) or – 4 (5 - y) • Now you have a common factor (5 –y) (3x– 4)

  13. 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 : ( + ) ( - ).

  14. Try These • 1. a2– 16 • 2. x2– 25 • 3. 4y2– 16 • 4. 9y2– 25 • 5. 3r2– 81 • 6. 2a2 + 16

  15. 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.

  16. Try These • 1. a2– 8a + 16 • 2. x2 + 10x + 25 • 3. 4y2 + 16y + 16 • 4. 9y2 + 30y + 25 • 5. 3r2– 18r + 27 • 6. 2a2 + 8a - 8

  17. 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.

  18. X-Box Method (3)(-10)= -30 x -5 3x 3x2 -15x 2 -15 -13 -10 2x +2 3x2-13x-10 = (x-5)(3x+2)

  19. 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.

  20. 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.

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