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UNIT 1  : ALGEBRA PREP

LESSON 4:  OPERATIONS WITH POLYNOMIALS

 

Addition and Subtraction:

 

Example 1: “+” sign preceding brackets – simply drop the brackets and collect like terms

 

a)  (3x2 – 2x + 5) + (5x2 – 3x – 6) = 3x2 – 2x + 5 + 5x2 – 3x – 6                              ** drop brackets

                                                     = 3x2 + 5x2 –2x – 3x +5 – 6                                 ** Collect like terms

                                                     = 8x2 – 5x – 1

 

Example 2: “-” sign preceding brackets – multiply each term in the bracket by “-1” and collect like terms

 

(5x2 – 3x + 6) – (2x2 – 7x + 8) = (5x2 – 3x + 6) – 1(2x2 – 7x + 8)                            ** multiply 2nd bracket by “-1”

                                                = 5x2 – 3x + 6 – 2x2 + 7x - 8

                                                = 5x2 – 2x2 – 3x + 7x + 6 – 8                                    ** collect like terms

                                                = 3x2 + 4x - 2

 

Example 3:

 

a)  (7a2 – 3ab – 5b2) + ( 4a2 – 8ab – 2b2) = 7a2 – 3ab – 5b2 + 4a2 – 8ab – 2b2

                                                                = 7a2 + 4a2 – 3ab – 8ab – 5b2 – 2b2

                                                                = 11a2 – 11ab – 7b2

b) (8x + 3y – 5xy) – (4x – 7y – 6xy) = (8x + 3y – 5xy) – 1(4x – 7y – 6xy)              ** multiply 2nd bracket by “-1”

                                                         = 8x + 3y – 5xy – 4x + 7y + 6xy

                                                         = 8x – 4x + 3y + 7y – 5xy + 6xy                                  ** collect like terms

                                                         = 4x + 10y + xy

 

Multiplying with Polynomials (Expanding):

 

Example 1: Monomial x Polynomial – multiply each term in bracket by the monomial

 

a) –3x(2x2 – 5x + 7) = -3x(2x2) – 3x(-5x) – 3x(7)                                                      ** multiply each term by “-3x”

                                = -6x3 + 15x2 – 21x

b)  2x(5x – 3) –5(2x + 7) = 10x2 – 6x – 10x – 35                                                      ** multiply each term in 1st bracket by “2x” and 2nd bracket by “-5”

                                        = 10x2 – 16x – 35                                                               ** collect like terms

 

Example 2: Polynomial x Polynomial – multiply each term in 1st  bracket by each term in 2nd  bracket

 

a) (3x + 5)(2x – 7) = 3x(2x – 7) + 5(2x – 7)                                                               ** multiply each term in 1st  bracket by each term in 2nd  bracket

                              = 6x2 – 21x + 10x –35                                                                  ** expand as in previous example

b) (2x + 3)(3x2 – 5x – 2) = 2x(3x2 – 5x – 2) + 3(3x2 – 5x – 2)                                               ** multiply each term in 1st  bracket by each term in 2nd  bracket

                                       = 6x3 –10x2 – 4x + 9x2 –15x – 6                                          ** expand

                                       = 6x3 – x2 –19x – 6                                                              ** collect like terms

c) 2(5x – 3)2 – 3(2x – 1)(3x + 2) = 2(5x – 3)(5x – 3) – 3(2x – 1)(3x + 2)

                                                   = 2(25x2 – 15x – 15x + 9) – 3(6x2 + 4x – 3x – 2)

                                                   = 50x2 – 30x – 30x + 18 – 18x2 – 12x + 9x + 6

                                                   = 32x263x  + 24

 

 

 Factoring Polynomials

 

Review of Basic Factoring methods:

 

1. Common Factoring:

Factor 6x3 – 15x

Solution:  6x3 – 15x = 3x(2x2 – 5)                                          ** Find the HIGHEST COMMON FACTOR for each term ---  “3x”

                                                                                                ** Divide “3x” into each term to get the second factor  --- “2x2 – 5”

                                                                                                ** Check by expanding

2. Difference of Squares: Formula --  a2 – b2 = (a – b)(a + b)

Factor 49x2 – 64y2

Solution:  49x2 – 64y2 = (7x – 8y)(7x + 8y)

 

Factor x2 – 9y2

Solution: x2 – 9y2 = (x – 3y)(x + 3y)

 

3. Simple Trinomials: Form  x2 + bx + c  [Coefficient of x2 is 1]

a) Factor x2 + 5x + 6

 

Solution: Recall  x2 + 5x + 6 = (x + __ )(x + __ )                  ** We need two numbers that multiply to +6 and add to +5

                                                                                                ** Check all pairs of factors of 6:       {1, 6}  adds to 7

                                                                                                                                                            {2, 3}  adds to 5

               Hence  x2 + 5x + 6 = (x + 3 )(x + 2)                        ** Check by expanding

 

b) Factor x2 – 2x – 35

 

Solution: Recall  x2 - 2x - 35 = (x + __ )(x + __ )                  ** We need two numbers that multiply to -35 and add to -2

                                                                                                ** Check all pairs of factors of 35:     {1, 35}  no combination adds to “-2”

                                                                                                                                                            {5, 7}  -7 + 5 = -2

               Hence  x2 – 2x - 35 = (x + 5 )(x – 7)                                    ** Check by expanding

 

4. Hard Trinomials: Form  ax2 + bx + c  [Coefficient of x2  does not equal 1]

a) Factor 6m2 – 5m – 4

Solution:

We use the method of decomposition (although there are other methods)

We decompose the middle term “-5m” into two parts using the two clues:

            Multiply to (6)(-4) = -24   and

            Add to “-5”

Strategy: List all pairs of factors of “-24” and find the pair that adds to “-5”          ** {1, 24}  cannot obtain “-5” with these two factors; 1+24=25; 1-24= -23

                                                                                                                        **  {2, 12} cannot obtain “-5” with these two factors

                                                                                                                        **  {3, 8}  -8 + 3 = -5 Choose these two factors

Hence  6m2 – 5m – 4 = 6m2 – 8m + 3m –4                                                     ** “-5m” broken into two parts “-8m + 3m”

                                    = 2m(3m – 4) + 1(3m – 4)                                           ** Group by twos and common factor

                                    = (3m – 4)(2m + 1)                                                      ** check by expanding                                                                                                          

b) Factor 2x2 – 9x + 4

Solution:

We decompose the middle term “-9x” into two parts using the two clues:

            Multiply to (2)(4) = +8   and

            Add to “-9”

Strategy: List all pairs of factors of “+8” and find the pair that adds to “-9”           ** {2, 4}  cannot obtain “-9” with these two factors; 2+4=6; 2-4= -2

                                                                                                                        ** {1, 8}  -8 + -1 = -9  Choose these two factors

Hence  2x2 – 9x + 4 = 2x2 –8x – 1x + 4                                                          ** “-9x” broken into two parts “-8x – 1x”

                                    = 2x(x – 4) – 1(x – 4)                                                  ** Group by twos and common factor

                                    = (x – 4)(2x – 1)                                                          ** check by expanding          

 

 

 

 

 

           

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