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Real Numbers

Radicals

Linear Equations

Inequalities & Absolute Value

Polynomials

Rational Expressions 1

Rat'l. Exp. 2 - mult&div

Rat'l. Exp. 3 - add&subt

Exponents

Slope & Equations of Lines

Review&Test


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

= 32x2 63x + 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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