C. D ETERMINING F ACTORS AND P RODUCTS Math 10: Foundations and Pre-Calculus

FP10.1 Demonstrate understanding of factors of whole numbers by determining the: prime factors greatest common factor least common multiple principal square root cube root. FP10.5 Demonstrate understanding of the multiplication and factoring of polynomial expressions (concretely, pictorially, and symbolically) including: multiplying of monomials, binomials, and trinomials common factors trinomial factoring relating multiplication and factoring of polynomials.

K EY TERMS : Find the definition of each of the following terms: Prime Factorization Greatest Common Factor Least Common Multiple Perfect Cube Cube Root Radicand Radical Index Factoring by Decomposition Perfect Square Trinomial Difference of Squares

1. F ACTORS AND M ULTIPLES OF W HOLE N UMBERS FP10.1 Demonstrate understanding of factors of whole numbers by determining the: prime factors greatest common factor least common multiple

1. F ACTORS AND M ULTIPLES OF W HOLE N UMBERS Two belts are created, one 12 beads long and the second 40 beads long. How many beads long must a belt be for it to created using either pattern?

Construct Understanding p.134

When a factor of a number has exactly 2 divisors, 1 and itself, the factor is a Prime Factor For example, the factors of 12 are 1,2,3,4,6,12. The prime factors are 2 and 3. To determine the prime factorization of 12, write as a product of its prime factors.

To avoid confusing the multiplication sign with variable x, we use a dot to represent the multiplication operation. Example

The 1 st 10 prime numbers are: 2,3,5,7,11,13,17,19,23,29 Natural numbers greater than 1 that are not prime are composite

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For 2 or more natural numbers, we can determine their Greatest Common Factor.

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To generate multiples of a number, multiply the number by the natural numbers, that is 1,2,3,4,5, etc. For example lets find the multiples of 26.

For 2 or more natural numbers, we can determine their Lowest Common Multiple When producing multiples for each number the first common one that comes up is the LCM.

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P RACTICE Ex. 3.1 (p. 140) #3-20

2. P ERFECT S QUARES, C UBES AND T HEIR R OOTS FP10.1 Demonstrate understanding of factors of whole numbers by determining the: principal square root cube root.

2. P ERFECT S QUARES, C UBES AND T HEIR R OOTS

So a perfect cube is a number that can be written as an integer multiplied by itself three times Example – 8 is a perfect cube because….

Which number from 1 to 200 represent perfect squares? Which represent perfect cubes?

Recall that a perfect square is a number that can be written as an integer multiplied by itself Example – 36 is a perfect square because….

Any whole number that can be represented as the area of a square with a whole number side length is a perfect square The side length of the square is the square root of the area of the square. Example

Any whole number that can be represented can be represented as the volume of a cube with a whole number edge length is a perfect cube The edge length of the cube is the cube root of the volume of the cube. Example

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P RACTICE Ex. 3.2 (p. 14) #1-14, 17 #1-5, 7-18

3. F ACTORS IN P OLYNOMIALS FP10.5 Demonstrate understanding of the multiplication and factoring of polynomial expressions (concretely, pictorially, and symbolically) including: multiplying of monomials, binomials, and trinomials common factors

3. F ACTORS IN P OLYNOMIALS

When we write a polynomial as a product of factors, we factor the polynomial. The diagrams on the last slide show that there are 3 ways to factor the expression 4m+12

Lets compare multiplying and factoring in arithmetic and algebra

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P RACTICE Ex. 3.3 (p. 154) #3-18 #3-5, 9-21

4. P OLYNOMIALS OF THE FORM X 2 + BX + C FP10.5 Demonstrate understanding of the multiplication and factoring of polynomial expressions (concretely, pictorially, and symbolically) including: multiplying of monomials, binomials, and trinomials common factors trinomial factoring relating multiplication and factoring of polynomials.

4. P OLYNOMIALS OF THE FORM X 2 + BX + C

Construct Understanding p. 159 Pattern – coefficient of c 2 is the product of the coefficients in front of c terms - End (lone) coefficient is the product of the two lone coefficients - The coefficient in front of c term is the sum of the two lone coefficients

Strategies – FOIL, Algebra Tiles, Area Model

When 2 binomials contain only positive terms, here are 2 strategies that can be used to determine the product of the binomials. Algebra Tiles (c+5)(c+3) Area Model (h+11)(h+5)

These strategies show that there are 4 terms in the product These terms are formed by applying distributive property and multiplying each term in the first binomial by each term in the second binomial. Example (h+11)(h+5)

The acronym, and method used to remember the multiplying strategy is FOIL. F – 1 st term in each mult. together O – outside term in each mult. together I – inside terms in each mult. together L – last term in each mult. together Then we add like terms together

When binomials have negative terms it is hard to use algebra tiles or the area model so we use FOIL to solve.

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Factoring and multiplying are inverses of each other. We can use this to factor a trinomial.

We can factor using another method which we will look at with an example. Factor z 2 -12z+35

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We should always check our factors by expanding back out. If we get back what we factored our factors are correct. Lets check example b.

The trinomials in ex 2 are written is descending order, that is, the terms are written in order from the term with the greatest exponent first to the term with the least exponent. When the order of the terms are reversed, the terms are written in ascending order.

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A trinomial that can be written as the product of two binomial factors may also have a common factor.

R ULE : Also look to see if there is a common factor that can be taken out of all three terms 1 st. Then factors into 2 binomials.

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When we compare factors of a trinomial, it is important to remember that the order in which we add terms does not matter x+a=a+x, for any integer a

Similarly, the order in which we multiply terms does not matter. (x+a)(x+b)=(x+b)(x+a)

P RACTICE Ex. 3.5 (p. 165) #2-21 #2-3, 8-23

5. P OLYNOMIALS OF THE FORM A X 2 + BX + C FP10.5 Demonstrate understanding of the multiplication and factoring of polynomial expressions (concretely, pictorially, and symbolically) including: multiplying of monomials, binomials, and trinomials common factors trinomial factoring relating multiplication and factoring of polynomials.

5. P OLYNOMIALS OF THE FORM A X 2 + BX + C

To multiply two binomials where the coefficients of the variables are not 1, we can use the same strategies as before.

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When binomials contain negative terms, it can be difficult to model their product with tiles. Using FOIL is then the best method!

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The best method for factoring a polynomial of the form ax 2 +bx+c is by using a method called decomposition Consider the binomial product: (3h+4)(2h+1) We can use FOIL

To factor 6h 2 +11h+4 by decomposition, we reverse the steps above. Notice that the coefficients of the h-terms have the product 3(8)=24 This is equal to the product of the coefficients of the h 2 -term and the constant term: 6(4)=24

So, to factor 6h 2 +11h+4, we decompose the h- term and write is as a sum of 2 terms whose coefficients have a product of 24. Back to our example.

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To check that the factors are correct, multiply them. Check example 3b.

Since this trinomial is the same as the original trinomial, the factors are correct.

P RACTICE Ex. 3.6 (p. 176) #1-3, 5-20 #1-3, 8-23

6. M ULTIPLYING P OLYNOMIALS FP10.5 Demonstrate understanding of the multiplication and factoring of polynomial expressions (concretely, pictorially, and symbolically) including: multiplying of monomials, binomials, and trinomials relating multiplication and factoring of polynomials.

6. M ULTIPLYING P OLYNOMIALS Construct Understanding p. 182

The distributive property can be used to perform any polynomial multiplication Each term of one polynomial must be multiplied by each term in the other polynomial.

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One way to check that your product is correct is to substitute a number in for the variable in both the product statement and your simplified answer. If both expressions are equal, your product in likely correct.

Lets check example 1a

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P RACTICE Ex. 3.7 (p. 185) #1-17 #1-3, 6-22

7. S PECIAL P OLYNOMIALS FP10.5 Demonstrate understanding of the multiplication and factoring of polynomial expressions (concretely, pictorially, and symbolically) including: multiplying of monomials, binomials, and trinomials common factors trinomial factoring relating multiplication and factoring of polynomials.

7. S PECIAL P OLYNOMIALS Construct Understanding p. 188

We say that a 2 +ab+b 2 is a perfect square trinomial because the 1 st and 3 rd terms are perfect squares and the middle term is twice the product of their square roots Example

In the factored form a perfect square trinomial is: We can use these patterns to factor perfect square trinomials

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If the trinomial is not a perfect trinomial square we have to factor, straight factoring or decomposition.

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Another example of a special polynomial is a difference of squares A difference of squares is a binomial of the form a 2 – b 2 We can think of it as a trinomial with a middle term of 0, That is, write a 2 – b 2 as a 2 + 0ab – b 2

Lets work through x 2 – 25

Short cut to factoring a difference of squares

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P RACTICE Ex. 3.8 (p.194) #2, 4-18 #2, 7-21