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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 18: FACTORING

 

SECTION 18.1: FACTORING GCF FROM A POLYNOMIAL

A greatest common factor(GCF) is the "largest" factor common to a number of terms in a polynomial. Here is the procedure for finding the GCF.

HOW TO FIND THE GCF OF A GIVEN POLYNOMIAL

 

Step 1) Rewrite each term as "factors of primes."

Step 2) Determine the factors, BOTH NUMERICAL AND VARIABLE, that are common to all of the terms in a given polynomial.

Comment: Often there will be more than one factor common to all the terms.

Step 3) Multiply all the common factors together. This will produce the GCF.

Step 4) Multiply the GCF found in step 1) by the polynomial that will result in the original polynomial

 

Example 1) Factor the GCF from $2x^{2}+4x+24$

Step 1)
2x2= (2)(x)(x)

4x = (2)(2)(x)

24 = (8)(3) = (2)(2)(2)(3)

Step 2 and 3)
The GCF is 2:
Why? 2 is the largest factor common to each term

Step 4) 2(x$^{2}$ + 2x + 12)

Check: MATH

Check is correct!

Example 2) Factor the GCF from 18x$^{2}$ + 27x + 81

Step 1)

18x2= (2)(3)(3)(x)(x)

27x = (3)(3)(3)(x)

81 = (3)(3)(3)(3)

Step 2 and 3) The GCF is 9:
Why? There are PAIRS of 3's in common WITH EACH TERM and I multiplied the 3's together.

Step 4) 9(2x$^{2}$ + 3x + 9)

Check: MATH   check is correct!

 

Example 3) Factor the GCF from MATH

Step 1)

MATH

MATH

MATH

Step 2) The GCF is 9x:
Why? There are pairs of 3's in common WITH EACH TERM as well as one x in common with each term. I multiplied the pair of 3's and the x together

Step 3) 9x(2x$^{2}$ + 3x + 9)

Check: MATH check is correct!

KEYS TO UNLOCKING THE GCF

Key 1) The numerical GCF is the LARGEST NUMBER that will divide WITHOUT REMAINDER the numerical parts of EVERY TERM.

Key 2) The variable GCF will be the variable exponential with the lowest exponent; and the variable exponential's base MUST BE COMMON TO ALL THE TERMS.

Key 3) The GCF will be the product of the numerical and variable GCF's.

Example 1) Factor the GCF from MATH

KEY 1) 30 divides 30, 60 and 90 of EACH TERM without remainder!

KEY 2) x$^{2}$ is the lowest variable exponential and x is COMMON to all terms.

KEY 3) The GCF is 30x2

answer: MATH

 

Example 2) Factor the GCF from MATH

KEY 1) 8 divides 16, 8 and 24 of EACH TERM without remainder!

KEY 2) x10 is the lowest variable exponential and x is COMMON to all terms.

KEY 3) The GCF is 8x10

answer: MATH

 

Example 3) Factor the GCF from MATH

KEY 1) 8 divides 16, 8 and 24 of EACH TERM without remainder!

KEY 2) x13 is the lowest variable exponential, but x is NOT IN COMMON to all terms. Therefore there is NO variable GCF!

KEY 3) The GCF is 8.

answer: MATH

 

Example 4) Factor the GCF from $16x^{16}+8x^{13}+1$

KEY 1) 8 divides 16 and 8 without remainder; but 8 WILL NOT DIVIDE 1 without remainder! Therefore there is no numerical GCF.

KEY 2) x13 is the lowest variable exponential but x is NOT IN COMMON to all terms. Therefore there is NO variable GCF!

KEY 3) The is NO GCF!

answer: $16x^{16}+8x^{13}+1$

 

SECTION 18.2: FACTORING TRINOMIALS, IN THE FORM OF ax2 + bx + c, where a, b, and c are
integers and a = +1, INTO TWO BINOMIALS

You learned how to multiply binomials like (x + 3)(x +2). The result of this multiplication is x2 + 5x + 6.

Now what if I asked you "what 2 binomials would you have to multiply together to get the trinomial x2+ 5x + 6?" You are being asked to think in "reverse."

FACTORING A TRINOMIAL INTO TWO BINOMIALS
MEANS


REWRITE THE TRINOMIAL AS TWO
BINOMIALS MULTIPLIED TOGETHER.

Let's go through an example of factoring a trinomial, in the form of $ax^{2}+bx+c$ where a = +1 and b and c are integers, into two binomials.

I will show you a "recipe" for factoring!

Example 1) Factor $x^{2}+9x+18$

Comment: notice that the coefficient of x$^{2}$ is 1. For my trick to work x$^{2}$ must have 1 as a coefficient.

Step 1) Place a rectangle around the 9 and the operation in front of 9.

Think of the operation in front of the 9 as the "sign" of 9.

In this case +9.

Figure

Step 2) Place a rectangle around the 18 and the operation in front of 18.

Think of the operation in front of the 18 as the "sign" of 18. In this case +18.

Figure

Step 3) Find a pair of factors of +18 that when multiplied will produce +18 but when added will produce +9.

+ 6 and + 3. Why? (6)(3) = 18 and 6 + 3 = 9

Step 4) Your answer will be (x + 6)(x + 3)

Figure

comment: the order of the +6 and + 3 is irrelevant.

CHECK: (x + 6)(x + 3) = x(x) + x(3) + 6(x) + 6(3) = x$^{2}$ + 3x + 6x + 18 = x$^{2}$ + 9x + 18

 

Example 2) Factor x$^{2}$ + 13x + 30

Step 1) Place a rectangle around the 13 and the operation in front of 13.

Think of the operation in front of the 13 as the "sign" of 13.

In this case 13 is +13.

Figure

Step 2) Place a rectangle around the 30 and the operation in front of 30.

Think of the operation in front of the 30 as the "sign" of 30. In this case 30 is +30.

Figure

Step 3) Find a pair of factors of +30 that when multiplied will produce +30 but when added will produce +13.

+ 10 and + 3. Why? (10)(3) = 30 and 10 + 3 = 13

Step 4) Your answer will be (x + 10)(x + 3)

Figure

CHECK: (x + 10)(x + 3) = x(x) + x(3) + 10(x) + 10(3) = x$^{2}$ + 3x + 10x + 30 = x$^{2}$ + 13x + 30

 

Example 3) Factor x$^{2}$ - 3x - 28

Step 1) Place a rectangle around the 3 and the operation in front of 3.

Think of the operation in front of the 3 as the "sign" of 3.

In this case 3 is -3.

Figure

Step 2) Place a rectangle around the 28 and the operation in front of 28.

Think of the operation in front of the 28 as the "sign" of 28. In this case 28 is -28.

Figure

Step 3) Find a pair of factors of -28 that when multiplied will produce -28 but when added will produce -3.

- 7 and + 4. Why? (-7)(4) = -28 and -7 + 4 = -3

 

Step 4) Your answer will be (x - 7)(x + 4)

Figure

CHECK: (x - 7)(x + 4) = (x + -7)(x + 4) = x(x) + x(4) + -7(x) + -7(4) = x$^{2}$ + 4x + -7x + -28  = x$^{2}$ + -3x + -28  = x$^{2}$ - 3x - 28

SOME COMMENTS ON MULTIPLYING BINOMIALS


WHEN MULTIPLYING BINOMIALS BY BINOMIALS
I CHANGED ALL SUBTRACTION TO ADDITION.

YOU MUST START LEARNING TO DO THIS IN YOUR HEAD!

 

Example 4) Factor $x^{2}-10x+24$

I plan to skip some of the steps given the fact that I have done so many examples.


Figure

- 6 and - 4 will be the proper numbers. $(-6)(-4)=+24$ and $-6+-4=-10$

Your answer will be $(x-6)(x-4)$.

CHECK:

MATH

= $x^{2}+-4x+-6x+24$

= $x^{2}+-10x+24$

= $x^{2}-10x+24$


Figure

 

SECTION 18.3: FACTORING TRINOMIALS IN THE FORM OF, ax2 + bx + c, WHERE a, b, and c ARE INTEGERS AND a IS NOT EQUAL TO 1

Consider the following trinomial $10x^{2}-13x-3$. Notice the lead coefficient is NOT equal to 1. We MUST use a different technique to factor this trinomial. I believe the technique that I am introducing to you is the easiest way to factor.

Example) Factor $10x^{2}-13x-3$

Step 1) Factor the coefficient of x$^{2}$, in this case 10, into all the factor pairs that will result in a product of this coefficient: order is irrelevant. Do the same for the "numerical part, i.e. the number that does not have a variable factor", in this case 3, but also swap the order of the factors. DO NOT WORRY ABOUT THE SIGN OF 3.


Figure

Step 2) Pair up the group 1 factor pairs with the group 2 factor pairs.

(5)(2)   (3)(1)
(5)(2)   (1)(3)
(10)(1)   (3)(1)
(10)(1)   (1)(3)

HINT: IF YOU WANT TO KNOW HOW MANY PAIRINGS YOU WILL HAVE MULTIPLY
THE NUMBER OF PAIRS IN THE FIRST GROUP, IN THIS CASE 2, BY THE
NUMBER OF PAIRS IN THE SECOND GROUP, ALSO TWO.
SO THE TOTAL NUMBER OF FACTOR PAIRS WILL BE 2 x 2 = 4.

Step 3) You will multiply the outer and inner factor pairs producing two numbers called the "result pair." See the image below:


Figure

Step 4) Look at the OPERATION of the third term of your trinomial you are factoring. If the third term is subtraction you must find a result pair that subtracts - subtract smaller number from the larger - to the coefficient of the second term. If the third term is addition you must find a result pair that adds to the coefficient of the second term.

factor1

Since 10x2 - 13 x - 3 has a third term, the numerical part, with an operation of subtraction, THE TRICK is that we must look for the result pair that will subtract to 13, the coefficient of of the second term.


Figure

YOU HAVE FOUND THE CORRECT FACTOR PAIR: (5)(2) and (1)(3). Notice how I DO NOT change the order of the factor pairs as they were written above - THIS IS IMPORTANT!

Step 5) The first factor pair, (5)(2), goes into the first position of each binomial in the original order the factor pairs were written; the second factor pair,(1)(3), goes into the second position of each binomial, in the original order the factor pairs were written.


Figure

The result being: (5     1)(2     3)

Step 6) Place the x's in the first position of each binomial. Then determine your operations in the binomials that will produce $10x^{2}-13x-3$ when the binomials are multiplied.

(5x       1)(2x       3)

and

(5x + 1)(2x - 3) will produce 10x2 - 13x - 3

Therefore, MATH factors into the two binomials MATH

Check: MATH

COMMENT: STUDENTS ALWAYS ASK ME HOW TO DETERMINE THE OPERATIONS THAT GO INTO THE BINOMIALS. FOR THE TIME BEING I WILL MAKE YOU FIGURE OUT THE OPERATIONS ON YOUR OWN. BUT THERE IS A "TRICK" TO DETERMINE THE OPERATIONS AS WELL.

example 1) Factor $12x^{2}-28x+15$


Figure


Figure

(6x      5)(2x      3 )

with operations

(6x - 5)(2x - 3 )

S.A.T. HINT: In general the factor pairs that have "large distances" between them, e.g. (12)(1) and (15)(1), will probably not be one of your answers. So in the above example you can initially "forget" about (12)(1) and (15)(1). BUT, I cannot guarantee that these factor pairs are not going to be in your final answer.

 

example 2) Factor $18x^{2}-57x+35$

group 1 factor pairs group 2 factor pairs
(6)(3) (7)(5)    (5)(7)
(9)(2) (35)(1)    (1)(35)
(18)(1)

I'll eliminate (18)(1) and (35)(1) and (1)(35) from my options. REMEMBER MY HINT: Big Gaps Between Factors Can Usually Be Weeded Out.

Since the operation of the third term is positive my inner and outer products must add to 57.

I believe (6)(3) and (5)(7) are the correct factor pairs.

Figure

(6 5)(3 7)

answer: (6x - 5)(3x - 7)

Again, it is up to you to determine the correct operations in the binomials.

 

SECTION 18.4: WHAT IS A PERFECT SQUARE?

A "perfect square" is a "positive integer" or a "math expression" that is the result of a positive integer squared or a math expression squared.


Figure

 

SECTION 18.5: WHAT IS THE DIFFERENCE OF TWO SQUARES?

What is similar about ex. 1 through ex. 3 below?

ex. 1) x2- 16

ex. 2) 4d4- 16

ex. 3) 9k2- 100m4

Explanation:

- Each of the above are made up of two terms, i.e. binomials, that are subtracted.

- The terms in each binomial are "perfect squares".

ex. 1) x2- 16 = (x)2 - (4)$^{2}$

ex. 2) 4d$^{2}$ - 16 = (2d)$^{2}$ - (4)$^{2}$

ex. 3) 9k$^{2}$ - 100m$^{4}$ = (3k)$^{2}$ - (10m$^{2}$)$^{2}$


Figure

 

SECTION 18.6: FACTORING THE DIFFERENCE OF TWO SQUARES


Figure

ex 1) MATH

ex 2) MATH

ex. 3) MATH

 

 

SECTION 18.7: WHAT IS A PERFECT CUBE?

A "perfect cube" is a "positive integer" or a "math expression" that is the result of a positive integer cubed or a math expression cubed.


Figure

 

 

SECTION 18.8: FACTORING THE SUM OR DIFFERENCE OF TWO PERFECT CUBES

Here are some examples of a "DIFFERENCE OF TWO CUBES."

ex. 1) $x^{3}-8$

ex. 2) $x^{3}-27$

ex. 3) $8x^{3}-1$

Notice that each term in the above examples is a perfect cube. Now I will write the second term in the above examples as a number cubed.

ex. 1) MATH

ex. 2) MATH

ex. 3) MATH

Here are some examples of a "SUM OF TWO CUBES." Now I will write the second term in the above examples as a number cubed.

ex. 1) $x^{3}+8$

ex. 2) $x^{3}+27$

ex. 3) $8x^{3}+1$

Notice that each term in the above examples is a perfect cube.

Now I will write the second term in the above examples as a number cubed.

ex. 1) MATH

ex. 2) MATH

ex. 3) MATH

 

There is a very simple pattern recognition rule for factoring a sum or a difference of two cubes.

RULES FOR FACTORING A SUM OR DIFFERENCE OF TWO CUBES

MATH

AND

MATH

 

Examples) Factor the following using the rule : MATH

ex. 1) $x^{3}-8$

step 1) $x^{3}-2^{3}$

a = x and b = 2

step 2) $(x-2)(x^{2}+2x+4)$

 

ex. 2) $x^{3}-27$

step 1) $x^{3}-3^{3}$

a = x and b = 3

step 2) $(x-3)(x^{2}+3x+9)$

ex. 3) $8x^{3}-1$

step 1) MATH

a = 2x and b = 1

step 2) MATH

Examples: Factor the following using the rule : MATH

ex. 1) $x^{3}+8$

step 1) $x^{3}+2^{3}$

a = x and b = 2

step 2) $(x+2)(x^{2}-2x+4)$

ex. 2) $x^{3}+27$

step 1) $x^{3}+3^{3}$

a = x and b = 3

step 2) $(x+3)(x^{2}-3x+9)$

 

ex. 3) $8x^{3}+1$

step 1) MATH

a = 2x and b = 1

step 2) MATH

MORE PROBLEMS OVER CHAPTER 18

Factor the following problems:

ex. 1) $21x+6x\U{b2}$   ans: $3x(2x+7)$
ex. 2) MATH   ans: MATH
ex. 3) $2x\U{b2}-14x+20$   ans: $(2x-4)(x-5)$
ex. 4) $x\U{b2}-2x-24$   ans: $(x+4)(x-6)$
ex. 5) $x\U{b2}+17x+70$   ans: $(x+7)(x+10)$
ex. 6) $2x\U{b2}-17x+21$   ans: $(2x-3)(x-7)$
ex. 7) $x\U{b2}-144$   ans: $(x-12)(x+12)$
ex. 8) $4x\U{b2}-49$   ans: $(2x-7)(2x+7) $
ex. 9) MATH   ans:MATH
ex. 10) MATH   ans: MATH
 

-----------------------------
THE SUMMER STUDY GUIDE
BY CHAPTERS

-----------------------------

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES