Common Factors of a Polynomial Polynomials Lesson 1 Common Factors of a Polynomial

Todays Objectives Students will be able to demonstrate an understanding of common factors and trinomial factoring, including: Determine the common factors in the terms of a polynomial, and express the polynomial in factored form Model the factoring of a trinomial, and record the process symbolically

Vocabulary Polynomial - one term or the sum of terms whose variables have whole number exponents; e.g. x² - 2x (x1/2 +10 is not a polynomial) Monomial - a polynomial with one term; e.g.20, 2x, -x² Binomial - a polynomial with two terms; e.g. 2x + 4, x² - 10 Trinomial - a polynomial with three terms; e.g. x² - 2x + 8 Algebra tiles - diagrams used to represent polynomial expressions Rectangle diagrams - diagrams which use rectangles to describe polynomials

Algebra Tiles In today’s lesson we will learn to use tools called algebra tiles that can be used to represent polynomial expressions. We will use 6 different algebra tiles:

An example of Algebra Tiles 1 = x2 = x = 1 = -x2 = -x = -1 Yellow = positive Red = negative 2x2 -7x +3 (Note: all the lines go all the way through the rectangle)

Algebra Tiles small squares – represents “1 or -1” (depending on color), side length of 1, area = 1 rectangles – represents “x or -x”, length of x, width of 1, area = x large squares – represents “x2 or –x2”, side length of x, area = x2 x 1 x x A=x2 1 A=1 1 A=x

Example How can we represent the binomial 4x + 12 using algebra tiles? Solution: there are several different ways that we can represent this binomial. Find all the possible ways that you can create a rectangle out of the tiles for the binomial 4x + 12: (4 rectangles, 12 small squares) 1(4x + 12) = 4x + 12 4(x + 3) = 4x + 12 2(2x + 6) = 4x + 12

Algebra Tiles The diagrams above show that there are three ways to factor the expression 4m + 12. The first two ways we say are incomplete because they can both be factored further. The third way we say is complete because the GCF of 4m and 12 is 4. Example) Factor the binomial 6n + 9 using algebra tiles Solution: The dimensions of the rectangle are 3 and 2n + 3. So, 6n + 9 = 3(2n + 3).

Example (You do) Factor the binomial 6c + 4c2 using algebra tiles and by finding the GCF Solution: Algebra Tiles Solution: Find the GCF 6𝑐=2∗3∗𝑐 4 𝑐 2 =2∗2∗𝑐∗𝑐 The GCF is 2c. Write each term as a product of 2c and another polynomial: 6𝑐+4 𝑐 2 =2𝑐(3+2𝑐)

Example Use algebra tiles to factor the binomial x2 – 1 Solution: (x-1)(x+1)

Example Try to factor the trinomial -5x2-10x+5 using algebra tiles (-x-1)(-5x-5) = 5x2+5x+5x+5 = 5x2+10x+5 doesn’t work

Algebra Tiles When a polynomial has negative terms or 3 different terms (a trinomial), we cannot remove a common factor by arranging the tiles as a rectangle. Instead, we can sometimes arrange the tiles into equal groups.

Example: Factoring Trinomials Factor the trinomial 5 – 10z – 5z2. Verify that the factors are correct. Solution: To factor a trinomial using algebra tiles, we arrange the tiles into equal groups instead of trying to make rectangles.

Example: Factoring Trinomials There are 5 equal groups and each group contains the trinomial 1 – 2z – z2. So, the factors are 5 and 1–2z–z2; 5–10z–5z2 = 5(1- 2z – z2) Another method is finding the GCF of the each term of the trinomial: 5 = 5 10z = 2*5*z 5z2 = 5*z*z The GCF is 5. So, 5 – 10z – 5z2 = 5(1 – 2z – z2)

Example (You do) Factor the trinomial. Verify that the factors are correct. -12x3y – 20xy2 – 16x2y2 Solution: Factor each term of the trinomial. 12x3y = 2*2*3*x*x*x*y 20xy2 = 2*2*5*x*y*y 16x2y2 = 2*2*2*2*x*x*y*y The GCF is 2*2*x*y = 4xy. Remove the GCF from each term. 4xy(-3x2 – 5y – 4xy)

Classroom work Pg. 155-156 # 4, 7, 9, 11, 14-18 If you finish, make sure your vocabulary books are up to date (finished chapter 3)