BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.

BMayer@ChabotCollege.edu • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

§5.2 Multiply§5.2 MultiplyPolyNomialsPolyNomials

Review §Review §

Any QUESTIONS About• §5.1 → PolyNomial Functions

Any QUESTIONS About HomeWork• §5.1 → HW-15

5.1 MTH 55

Multiply MonomialsMultiply Monomials Recall Monomial is a term that is a

product of constants and/or variables • Examples of monomials: 8, w, 24x3y

To Multiply MonomialsTo find an equivalent expression for the product of two monomials, multiply the coefficients and then multiply the variables using the product rule for exponents

From From §1.6§1.6 Exponent Properties Exponent Properties1 as an exponent a1 = a

0 as an exponent a0 = 1

Negative exponents

The Product RuleThe Product Rule

The Quotient Rule

The Power Rule (am)n = amn

Raising a product to a power

(ab)n = anbn

Raising a quotient to a power

.m n m na a a

n nn mn

a b a ba

b aa b a

This sum

mary assum

es that no denom

inators are 0 and that 00 is not

considered. For any integers m

Example Example Multiply Monomials Multiply Monomials

Multiply: a) (6x)(7x) b) (5a)(−a) c) (−8x6)(3x4)

Solution a) (6x)(7x) = (6 7) (x x) = 42x2

Solution b) (5a)(−a) = (5a)(−1a)

= (5)(−1)(a a) = −5a2

Solution c) (−8x6)(3x4) = (−8 3) (x6 x4)

= −24x6 + 4 = −24x10

(Monomial)•(Polynomial)(Monomial)•(Polynomial)

Recall that a polynomial is a monomial or a sum of monomials.• Examples of polynomials:

5w + 8, −3x2 + x + 4, x, 0, 75y6

Product of Monomial & Polynomial• To multiply a monomial and a polynomial,

multiply each term of the polynomial by the monomial.

Example Example (mono)•(poly) (mono)•(poly)

Multiply: a) x & x + 7 b) 6x(x2 − 4x + 5) Solution

a) x(x + 7) = x x + x 7

= x2 + 7x

b) 6x(x2 − 4x + 5) = (6x)(x2) − (6x)(4x) + (6x)(5)

= 6x3 − 24x2 + 30x

Example Example (mono)•(poly) (mono)•(poly)

Multiply: 5x2(x3 − 4x2 + 3x − 5)

Solution:

5x2(x3 − 4x2 + 3x − 5) =

= 5x5 − 20x4 + 15x3 − 25x2

Product of Two PolynomialsProduct of Two Polynomials

To multiply two polynomials, P and Q, select one of the polynomials, say P. Then multiply each term of P by every term of Q and combine like terms.

Example Example (poly)•(poly) (poly)•(poly)

Multiply x + 3 and x + 5

Solution (x + 3)(x + 5) = (x + 3)x + (x + 3)5

= x(x + 3) + 5(x + 3)

= x x + x 3 + 5 x + 5 3

= x2 + 3x + 5x + 15

= x2 + 8x + 15

Multiply 3x − 2 and x − 1

Solution (3x − 2)(x − 1) = (3x − 2)x − (3x − 2)1

= x(3x − 2) – 1(3x − 2)

= x 3x − x 2 − 1 3x − 1(−2) = 3x2 − 2x − 3x + 2

= 3x2 − 5x + 2

Multiply: (5x3 + x2 + 4x)(x2 + 3x)

Solution: 5x3 + x2 + 4x

x2 + 3x

15x4 + 3x3 + 12x2

5x5 + x4 + 4x3

5x5 + 16x4 + 7x3 + 12x2

Multiply: (−3x2 − 4)(2x2 − 3x + 1)

Solution

2x2 − 3x + 1

−3x2 − 4

−8x2 + 12x − 4

−6x4 + 9x3 − 3x2

−6x4 + 9x3 − 11x2 + 12x − 4

PolyNomial Mult. SummaryPolyNomial Mult. Summary Multiplication of

polynomials is an extension of the distributive property. When you multiply two polynomials you distribute each term of one polynomial to each term of the other polynomial.

We can multiply polynomials in a vertical format like we would multiply two numbers

(x – 3)(x – 2)x_________

+ 6 –2x+ 0–3xx2_________

x2 –5x + 6

PolyNomial Mult. By FOILPolyNomial Mult. By FOIL FOIL Method

FOIL Example

(x – 3)(x – 2) = x2 – 5x + 6x(x) + x(–2) + (–3)(x) + (–3)(–2) =

FOIL ExampleFOIL Example

Multiply (x + 4)(x2 + 3)

Solution

F O I L

(x + 4)(x2 + 3) = x3 + 3x + 4x2 + 12

= x3 + 4x2 + 3x + 12

The terms are rearranged in descending order for the final answer

FOIL applies to ANY set of TWO BiNomials,

Regardless of the BiNomial Degree

More FOIL ExamplesMore FOIL Examples

Multiply (5t3 + 4t)(2t2 − 1) Solution:

(5t3 + 4t)(2t2 − 1) = 10t5 − 5t3 + 8t3 − 4t

= 10t5 + 3t3 − 4t Multiply (4 − 3x)(8 − 5x3) Solution:

(4 − 3x)(8 − 5x3) = 32 − 20x3 − 24x + 15x4

= 32 − 24x − 20x3 + 15x4

Special ProductsSpecial Products

Some pairs of binomials have special products (multiplication results).

When multiplied, these pairs of binomials always follow the same pattern.

By learning to recognize these pairs of binomials, you can use their multiplication patterns to find the product more quickly & easily

Difference of Two SquaresDifference of Two Squares

One special pair of binomials is the sum of two numbers times the difference of the same two numbers.

Let’s look at the numbers x and 4. The sum of x and 4 can be written (x + 4). The difference of x and 4 can be written (x − 4). The Product by FOIL:

(x + 4)(x – 4) = x2 – 4x + 4x – 16 = x2 – 16( )

Difference of Two SquaresDifference of Two Squares

Some More Examples

(x + 4)(x – 4) = x2 – 4x + 4x – 16 = x2 – 16

(x + 3)(x – 3) = x2 – 3x + 3x – 9 = x2 – 9

(5 – y)(5 + y) = 25 +5y – 5y – y2 = 25 – y2}What do all

of these have

in common?

ALL the Results are Difference of 2-Sqs:Formula → (A + B)(A – B) = A2 – B2

General Case F.O.I.L.General Case F.O.I.L.

Given the product of generic Linear Binomials (ax+b)·(cx+d) then FOILing:

Can be Combined IF BiNomials are LINEAR

Geometry of BiNomial MultGeometry of BiNomial Mult

The products oftwo binomials can be shown in terms of geometry; e.g,(x+7)·(x+5) →(Length)·(Width)

Width= (x+5)

Length= (x+7)

(Length)·(Width) = Sum of the areas of the four internal rectangles 7 5 x x 2 5 7 35 x x x

2 12 35 x x

Example Example Diff of Sqs Diff of Sqs

Multiply (x + 8)(x − 8) Solution: Recognize from Previous

Discussion that this formula Applies(A + B)(A − B) = A2 − B2

So (x + 8)(x − 8) = x2 − 82

= x2 − 64

Example Example Diff of Sqs Diff of Sqs

Multiply (6 + 5w)(6 − 5w) Solution: Again Diff of 2-Sqs

Applies → (A + B)(A − B) = A2 − B2

In this Case• A 6 & B 5w

So (6 + 5w) (6 − 5w) = 62 − (5w)2

= 36 − 25w2

Square of a BiNomialSquare of a BiNomial

The square of a binomial is the square of the first term, plus twice the product of the two terms, plus the square of the last term.

(A + B)2 = A2 + 2AB + B2

(A − B)2 = A2 − 2AB + B2

These are called perfect-square trinomials

222222: BABABABA NOTE

Example Example Sq of BiNomial Sq of BiNomial

Find: (x + 8)2

Solution: Use (A + B)2 = A2+2AB + B2

(x + 8)2 = x2 + 2x8 + 82

= x2 + 16x + 64

Example Example Sq of BiNomial Sq of BiNomial

Find: (4x − 3x5)2

Solution: Use (A − B)2 = A2 − 2AB + B2

In this Case• A 4x & B 3x5

(4x − 3x5)2 = (4x)2 − 2 4x 3x5 + (3x5)2

= 16x2 − 24x6 + 9x10

Summary Summary Binomial Products Binomial Products

Useful Formulas for Several Special Products of Binomials:

For any two numbers A and B, (A − B)2 = A2 − 2AB + B2

For two numbers A and B, (A + B)2 = A2 + 2AB + B2

For any two numbers A and B, (A + B)(A − B) = A2 − B2.

Multiply Two POLYnomialsMultiply Two POLYnomials

1. Is the multiplication the product of a monomial and a polynomial? If so, multiply each term of the polynomial by the monomial.

2. Is the multiplication the product of two binomials? If so:

a) Is the product of the sum and difference of the same two terms? If so, use pattern(A + B)(A − B) = A2 − B2

Multiply Two POLYnomialsMultiply Two POLYnomials

2. Is the multiplication the product of Two binomials? If so:

b) Is the product the square of a binomial? If so, use the pattern (A + B)2 = A2 + 2AB + B2, or (A − B)2 = A2 − 2AB + B2

c) c) If neither (a) nor (b) applies, use FOIL

3. Is the multiplication the product of two polynomials other than those above? If so, multiply each term of one by every term of the other (use Vertical form).

Example Example Multiply PolyNoms Multiply PolyNoms

a) (x + 5)(x − 5) b) (w − 7)(w + 4)

c) (x + 9)(x + 9) d) 3x2(4x2 + x − 2)

e) (p + 2)(p2 + 3p – 2)

SOLUTION

(x + 5)(x − 5) = x2 − 25

(w − 7)(w + 4) = w2 + 4w − 7w − 28

= w2 − 3w − 28

Example Example Multiply PolyNoms Multiply PolyNoms

SOLUTION

c) (x + 9)(x + 9) = x2 + 18x + 81

d) 3x2(4x2 + x − 2) = 12x4 + 3x3 − 6x2

e) By columns

p2 + 3p − 2 p + 2

2p2 + 6p − 4 p3 + 3p2 − 2p p3 + 5p2 + 4p − 4

Function NotationFunction Notation

From the viewpoint of functions, if

f(x) = x2 + 6x + 9

g(x) = (x + 3)2

Then for any given input x, the outputs f(x) and g(x) above are identical.

We say that f and g represent the same function

Example Example ff((a a + + hh) ) − − ff((aa))

For functions f described by second degree polynomials, find and simplify notation like f(a + h) and f(a + h) − f(a)

Given f(x) = x2 + 3x + 2, find and simplify f(a+h) and simplify f(a+h) − f(a)

SOLUTION

f (a + h) = (a + h) 2 + 3(a + h) + 2

= a 2 + 2ah + h

2 + 3a + 3h + 2

Example Example ff((a a + + hh) ) − f(− f(aa))

Given f(x) = x2 + 3x + 2, find and simplify f(a+h) and simplify f(a+h) − f(a)

SOLUTION

f (a + h) − f (a) =[(a + h)2 + 3( a + h) + 2] − [a2 + 3a + 2]

= a 2 + 2ah + h 2 + 3a + 3h + 2 − a2 − 3a − 2

= 2ah + h2 + 3h

Multiply PolyNomials as FcnsMultiply PolyNomials as Fcns

Recall from the discussion of the Algebra of Functions The product of two functions, f·g, is found by

(f·g)(x) = [f(x)]·[g(x)]

This can (obviously) be applied to PolyNomial Functions

Example Example Fcn Multiplication Fcn Multiplication

Given PolyNomial Functions

( ) 3f x x 2( ) 6 8g x x x Then Find: (f·g)(x) and (f·g)(−3) SOLUTION

23 6 8x x x 3 2 26 8 3 18 24x x x x x

(f · g)(x) = f(x) · g(x)

3 23 10 24x x x

(f · g)(−3)

3 23 3 3 10 3 24 27 27 30 24

WhiteBoard WorkWhiteBoard Work

Problems From §5.2 Exercise Set• 30, 54, 82, 98b, 116, 118

PerfectSquareTrinomialByGeometry

All Done for TodayAll Done for Today

Remember FOIL By

BIG NOSEDiagram

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

AppendiAppendixx

srsrsr 22

Graph Graph yy = | = |xx||

Make T-tablex y = |x |

-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls

-3 -2 -1 0 1 2 3 4 5

M55_§JBerland_Graphs_0806.xls

BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.

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