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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 17: MULTIPLYING POLYNOMIALS

 

SECTION 17.1: SUBTRACTION IS A FORM OF ADDITION

In an earlier section you learned that subtraction is a form of addition. For example, 5 -7 is actually 5 + -7 which is equal to -2. If you wish, you can rewrite all subtraction as addition if it enables you to perform certain mathematical processes more efficiently. Many students have difficulties when subtraction is mixed with a signed numbers so making this switch from subtraction may be useful. But in order to master algebra and upper level mathematics you must internalize this process so that you don't have to explicitly makes these changes on your work paper.

For example, 3x - 5 can be rewritten as 3x + -5. This ability will be useful in this section.

THE RULE FOR WRITING SUBTRACTION AS ADDITION
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Change subtraction to addition, then change the sign of the number that

was to the immediate right of the subtraction symbol to its opposite.

In other words, a positive number would become negative and a

negative number would become positive.
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Examples

Rewrite the following algebraic expressions as addition.

a) $6x-11$ $\ \ \ \ ans:6x+-11$

b) $-2x^{2}--12x-15$ MATH

 

SECTION 17.2: MULTIPLYING MONOMIALS BY MONOMIALS

 

DEFINITIONS

Monomial: a number, a variable, or a number times one or more variables.

ex.1)  5     ex. 2) x    ex. 3) -5x     ex. 4) xyz     ex. 5) 11yxzm


Factor: a "math expressions" that is multiplied.

ex. 1) (2)(3)(5)   2, 3 and 5 are factors
ex. 2) 5xyz   5, x, y and z are factors
ex. 3) 7(x + 2)(x - 4)   7, (x + 2) and (x - 4) are factors

MR. L'S COMMENTS:
As you know by now factors can be reordered
and
WILL NOT change the result of multiplication.
ex) (7)(-3) = (-3)(7)

As an aside, you are suppose to know this same principle
can be applied to terms that are ADDED.
ex) 5 + -3 = -3 + 5

 

You know from a previous section what 5s4  means 5(s)(s)(s)(s) since the power applies to the number or variable immediately to the left of the power. In this case s is the "base." You also know that variables or numbers that are raised to a power are called "exponentials." So, s4  would be called an "exponential."

Let's consider what $(5s^{4})(3s^{3})$ means: MATH MATH

Note that I "expanded" $(5s^{4})(3s^{3})$ into factors. Because of the properties of multiplication, in this case the commutative property of multiplication,.I am able to move the factors around to suit my purposes.

Consider the following:

MATH

From the above examples I can infer the following rule for multiplying exponentials which in this case are monomials.

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RULE FOR MULTIPLYING EXPONENTIALS

MATH

where a, b, m and n are real numbers.

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Essentially what we are doing above is multiplying "monomials."

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MORE EXAMPLES OF MONOMIALS

ex. 1) $-6x^{2}$ is equal to -6xx which is made up of 3 factors.

ex. 6) $12x^{2}y^{5}$ is equal to 12xxyyyyy which is made up of 8 factors

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Examples of Multiplying Monomials:

ex. 1) MATH
comment: There are 3 factors in 35xy. WHY?

ex. 2) (-2t3)(-4t3) = (-2)(-4)(t3)(t3) = 8(ttt)(ttt) = 8t6 

comment: There are 7 factors in 8t6. WHY?

ex. 3) MATH
comment: Notice I used the rule for multiplying exponentials in this problem. WHY?
Also, there are 14 factors in -56g13. WHY?

ex. 4) MATHMATH
comment: Notice I used the rule for multiplying exponentials in this problem. WHY?
Also, there are 235 factors in (-3/10)h234. WHY?

 

SECTION 17.3: MULTIPLYING MONOMIALS BY BINOMIALS AND MORE

Consider the following problem: $h^{3}(2h^{2}+5h)$ The operation between h3  and $2h^{2}+5h+5$ is multiplication. Why? What we have here, $h^{3}(2h^{2}+5h)$ , is a monomial times a binomial. A "binomial" is two monomials that are added(or subtracted).

Now based on the distributive property I have to multiply k3  through each term of the binomial $2h^{2}+5h$.

MATH

Let's try another problem, again using the distributive property

ex) Multiply MATH.

MATH

MATH

MATH

MATH


Figure

ex.) MATH

Step 1) MATH)

Step 2) -8u7(-6u9 ) + -8u7(-2u12 ) = 48u16 + 16u19

Step 3) $48u^{16}+16u^{19}$
Comment: These terms cannot be added because they are not "like terms."

Therefore: MATH

!!!!! IMPORTANT !!!!!
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Ideally you are expected to change all subtraction to
addition in your head. Teachers expect that
you can do this. I am changing subtraction to addition
in order to show you the
algebraic rule that justifies
the symbol manipulations.

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Now what about the situation when you multiply a monomial by a "trinomial" which is 3 monomials added or when you multiply a monomial by more than 3 monomials. At this point we will define a "polynomial."

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POLYNOMIAL

A polynomial is one or more monomials that are added.
AND
You will use the same rule you use for multiplying monomials by

binomials when multiplying a monomial times a polynomial.

Comment: The above definition of a polynomial is NOT exact and an
over simplification. This will suffice for now.

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ex.) Multiply the monomial by the polynomial: MATH

MATH

MATH

MATH

Comment: Notice how I did not leave a negative number to the right of addition. Remember, addition followed by a negative is subtraction; and subtraction followed by a negative is addition.

 

SECTION 17.4: MULTIPLYING BINOMIALS BY BINOMIALS


BINOMIAL TIMES A BINOMIAL RULE
STEP 1) Change all subtraction to addition based on the mathematical fact that subtraction is really addition.
STEP 2) Multiply the first term of the first binomial by the second binomial using the distributive property.
STEP 3) Multiply the second term of the first binomial by the second binomial using the distributive property.
STEP 4) Add the results from step 2 and step 3, then simplfy like terms.


You have been taught in previous classes that the above technique for multiplying a binomial times a binomial is called the F.O.I.L. technique.

ex 1. Multiply $(2x-3)(-7x+8)$

Step 1) $(2x+-3)(-7x+8)$ Note how I changed all subtraction to addition.

Step 2) $2x(-7x+8)$ produces (2x)(-7x) = -14x2 and (2x)(8) = 16x.

Step 3) $-3(-7x+8)$ produces (-3)(-7x) = 21x and (-3)(8) = -24.

Step 4) MATH

 

ex 2. Multiply $(-4x+7)(-2x-5)$

Step 1) $(-4x+7)(-2x+-5)$ Note how I changed all subtraction to addition.

Step 2) $-4x(-2x+-5)$ produces MATH and MATH

Step 3) $7(-2x+-5)$ produces $7(-2x)=-14x\ $ and $(7)(-5)=-35$

Step 4) MATH = $8x^{2}+6x-35$

 

ex 3. Multiply $(-3x-2)(-5x-8)$

Step 1) $(-3x+-2)(-5x+-8)$ Note how I changed all subtraction to addition.

Step 2) $-3x(-5x+-8)$ produces MATH and $(-3x)(-8)=24x$

Step 3) $-2(-5x+-8)$ produces $(-2)(-5x)=10x\ $ and $(-2)(-8)=16$

Step 4) MATH = $15x^{2}+34x+16$

 

 

SECTION 17.5: MULTIPLYING POLYNOMIALS BY POLYNOMIALS

Would you know how to multiply MATH? You are being asked to multiply a polynomial times a polynomial. Here is the rule:

MULTIPLYING A POLYNOMIAL BY A POLYNOMIAL
STEP 1) Change all subtraction to addition based on the mathematical fact that subtraction is really addition.
STEP 2) Multiply each term of the first polynomial by the second polynomial using the distributive property.
STEP 3) Add the results from step 2 , then simplfy like terms.

 

ex. 1) Multiply the two polynomials: MATH

Step 1) MATH $\ \ $

 

Step 2)
MATH

MATH MATH

MATH

Step 3) MATH

MATH

MATH

Try this on your own:

MATH

 

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

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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