Factoring 6 Types of Factoring:
1. Common Factoring 2. Difference of Squares 3. Perfect Square Trinomials 4. Simple Trinomials
5. Factoring by Grouping 6. Decomposition 1. Common Factoring -Find whats common
-Put the common factor in front and everything left over in a bracket EX. 15x3y2-25xy4+5xy
=5xy (3x2y-5y3) 2. Difference of Squares
-find the square root of each number
-place the root of the first number at the brackets along with any variables -place the second root after the first with one positive and one negative sign EX. 4k2-9
=(2k+3)(2k-3)
3. Perfect Square Trinomials
-find the root of the first and last term and square it EX. 25y2-70y+49
=(5y-7)2
4. Simple Trinomials
-find two numbers that multiply to the last term and add to the middle term -place the variable and two numbers inside two sets of brackets
EX. x2-3x-40 =(x+5)(x-8)
5. Factoring by Grouping
-common factor the first two terms and then the last two terms -Put the two terms outside the brackets into one bracket
EX. 10x3-5x2+6x-3 =5x2(2x-1)+3(2x-1) =(2x-1)(5x2+3) 6. Decomposition
-multiply the first and last term to find the product -the sum is the middle term
-Find two number that multiply to the product and add to the sum and replace them with the middle number
-factor by grouping
EX. 10x2-11x-6 product=-60 sum=-11 =10x2-15x+4x-6
=5x(2x-3)+2(2x-3) =(2x-3)(5x+2)
Type How to recognize it
Common Factoring A variable, coefficient or both is common between every term
Difference of Squares 2 terms, subtracted, both perfect squares Perfect Square Trinomials 3 terms, two end terms are perfect
squares, middle term is double the product
Simple Trinomials 3 terms, the first term has NO
coefficient (ie. No number in front) Factoring by Grouping 4 or more terms
Decomposition 3 terms, first term HAS a coefficient in front (ie does have a number in front) **Common factor first always**
Polynomials
Multiplying Polynomials
-To solve FOIL or use double distribution
bracket.
-Then collect like terms. EX.1 (2x-3)(x+5)
=2x2+10x-3x-15 =2x2+7x-15
EX.2 (5m2+3m-2)(2m3+7m2-m+5)
=10m5+35m4-5m3+25m2+6m4+21m3-3m2+15m-4m3-14m2+2m-10 =10m5+41m4+12m3+8m2+17m-10
**REMEBER EXPONENT LAWS: ADD YOUR EXPONENTS SINCE YOUR MULTIPLYING POWERS WITH THE SAME BASE**
Squaring Binomials (a+/-b)2 -square the 1st
-square the last -double the product EX. (3x-4y)2
=9x2-12xy-12xy+16y2 =9x2-24xy+16y2
Multiplying Identical Binomials (a-b)(a+b) -multiply first and last terms
-middle terms cancel out EX. (5x+2y)(5x-2y)
=25x2-10xy+10xy-4y2 =25x2-4y2
The exponent of a number says how many times to use the number in a multiplication.
EX. 82 = 8 × 8 = 64
Two powers with the SAME base
-If you have (x5)(x6); You are counting factors.
X5 = (x)(x)(x)(x)(x) and x6 =(x)(x)(x)(x)(x)(x) -So if you multiply them together:
(x5)(x6) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x11
-Five x factors from x5, and six x factors from x6, makes 11 x factors total. **When bases are the same you add exponents**
Two powers with DIFFERENT bases
-If you write out the powers, you see there’s no way you can combine them. (x3)(y4)= (x)(x)(x)(y)(y)(y)(y)
**If the bases are different but the exponents are the same, then you can combine them. **
EX. (x³)(y³) = (x)(x)(x)(y)(y)(y) = (xy)(xy)(xy) Which can be written as (xy)³.
Exponent Rules: Power of product, Power of quotient, Power of a power, The negative exponent, The zero exponent
1. Factor numerators and denominators as far as possible 2. Cancel common factors
3. Multiply numerators then denominators 4. Reduce anwser
5. State Restrictions EX 1.
2a3 X 3bc3 Cancel Factors Diagonally across equation 5b3c 8a3 (bold with bold, italic with italic)
= 1 X 3c2 multiply through! 5b2 4
= 3c2 20b2 Restrictions:
5b3c can’t equal 0 8a3 can’t equal 0 a, b, c can’t equal 0
Ex 2. (with factoring) 2x + 4 X 10x -10 5x - 5 7x + 14
= 2(x + 2) X 10(x - 1) Cancel Factors 5(x - 1) 7(x + 2)
= 2 X 2 4 1 7 = 7 Restrictions:
x can’t equal 1 x can’t equal -2
**When Dividing Rational Expressions, restrictions occur in numerators and denominators**
-Rational expressions are added and subtracted the same way as rational numbers are added and subtracted.
EX. M+( )
2
n m
m -n
3
= m n n
n n m m ) ( ) (
+ m n n
n m ) ( 2
- m n n
n m ) ( ) ( 3
= m n n
mn nm ) ( 2 ^ 2 ^
+ m n n
mn
) (
2
- m n n
n m ) ( 3 3
= m n n
n m mn mn nm ) ( 3 3 2 2 ^ 2 ^
Now state the restrictions: R: (m+n)n ± 0
m+n=0 n=0
m=-n
**If there is no common base, find a common base by multiplying each denominator together and looking for simpler common factors.**
Steps for Adding
1. Change fractions into equivalent fractions with a common denominator. 2. Add numerator, keep the common denominator.
3. Simplify the resulting expression, canceling common factor. Steps for Subtracting
1. Change fractions into equivalent fractions with a common denominator. 2. Subtract numerator, keep the common denominator.
3. Simplify the resulting expression, canceling common factor.
Simplifying Rational Expressions
-A rational expression, is a quotient, however it is a quotient of two polynomial expressions.
The following are examples of rational expression:
1 y+6 x 8-x
Now we are going to reduce a rational expression to its lowest terms
25x²y Restrictions: These occur because the denominator can’t 5xy be 0 always check the original denominator or in factored form.
