Multiplying Polynomials: Monomials, Binomials, and Special Products, Slides of Algebra

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§5.2 Multiply

PolyNomials

Review §

 Any QUESTIONS About

• §5.1 → PolyNomial Functions

 Any QUESTIONS About HomeWork

• §5.1 → HW-

5.1 MTH 55

From §1.6  Exponent Properties

1 as an exponent a^1 = a

0 as an exponent a^0 = 1

Negative exponents

The Product Rule

The Quotient Rule

The Power Rule ( a m ) n^ = a mn Raising a product to a power

( ab ) n^ = a n^ b n

Raising a quotient to a power.

n (^) n n

a a b (^) b

  (^) =  

.

m (^) m n n a (^) a a =^ −

a m^ ⋅ a n^ = a m^ + n.

n^1 , n^ m ,^ n^ n n m n a a^ b^ a^ b a b a b^ a

− −^ − = (^) − = ^ ^ =^ 

denominators are 0 and that 0 This summary assumes that no

is not^0

considered. For any integers

(^) m (^) and (^) n

Example  Multiply Monomials

• Multiply: a) (6 x )(7 x ) b) (5 a )(− a )

c) (− 8 x^6 )(3 x^4 )

 Solution a) (6 x )(7 x ) = (6 ⋅ 7) ( x ⋅ x ) = 42 x^2

 Solution b) (5 a )(− a ) = (5 a )(− 1 a )

= (5)(−1)( a ⋅ a ) = − 5 a^2

 Solution c) (− 8 x^6 )(3 x^4 ) = (− 8 ⋅ 3) ( x^6 ⋅ x^4 )

= − 24 x 6 + 4^ = − 24 x^10

Example  (mono)•(poly)

• Multiply: a) x & x + 7 b) 6 x ( x^2 − 4 x + 5)
• Solution

a) x ( x + 7) = x ⋅ x + x ⋅ 7 = x^2 + 7 x

b) 6 x ( x^2 − 4 x + 5) = (6 x )( x^2 ) − (6 x )(4 x ) + (6 x )(5)

= 6 x^3 − 24 x^2 + 30 x

Example  (mono)•(poly)

• Multiply: 5 x^2 ( x^3 − 4 x^2 + 3 x − 5)
• Solution :

5 x^2 ( x^3 − 4 x^2 + 3 x − 5) =

= 5 x^5 − 20 x^4 + 15 x^3 − 25 x^2

Example  (poly)•(poly)

• Multiply x + 3 and x + 5
• Solution ( x + 3)( x + 5) = ( x + 3) x + ( x + 3) = x ( x + 3) + 5( x + 3) = x ⋅ x + x ⋅ 3 + 5 ⋅ x + 5 ⋅ 3 = x^2 + 3 x + 5 x + 15 = x^2 + 8 x + 15

Example  (poly)•(poly)

• Multiply 3 x − 2 and x − 1
• Solution (3 x − 2)( x − 1) = (3 x − 2) x − (3 x − 2) = x (3 x − 2) – 1(3 x − 2) = x ⋅ 3 x − x ⋅ 2 − 1 ⋅ 3 x − 1(−2) = 3 x^2 − 2 x − 3 x + 2 = 3 x^2 − 5 x + 2

Example  (poly)•(poly)

 Multiply: (− 3 x^2 − 4)(2 x^2 − 3 x + 1)

 Solution

2 x^2 − 3 x + 1 − 3 x^2 − 4 − 8 x^2 + 12 x − 4 − 6 x^4 + 9 x^3 − 3 x^2 − 6 x^4 + 9 x^3 − 11 x^2 + 12 x − 4

PolyNomial 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_________^ (x – 2) –2x + 6 _________x^2 –3x + 0 x^2 –5x + 6

FOIL Example

• Multiply ( x + 4)( x^2 + 3)
• Solution

F O I L ( x + 4)( x^2 + 3) = x^3 + 3 x + 4 x^2 + 12

O

I

F L

= x^3 + 4 x^2 + 3 x + 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 Examples

• Multiply (5 t^3 + 4 t )(2 t^2 − 1)
• Solution : (5 t^3 + 4 t )(2 t^2 − 1) = 10 t^5 − 5 t^3 + 8 t^3 − 4 t = 10 t^5 + 3 t^3 − 4 t
• Multiply (4 − 3 x )(8 − 5 x^3 )
• Solution : (4 − 3 x )(8 − 5 x^3 ) = 32 − 20 x^3 − 24 x + 15 x^4 = 32 − 24 x − 20 x^3 + 15 x^4

Difference 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) = x^2 ( – 4x + 4x – 16 =^ ) x^2 – 16

Difference of Two Squares

• Some More Examples

(x + 4)(x – 4) = x^2 – 4x + 4x – 16 = x^2 – 16

(x + 3)(x – 3) = x^2 – 3x + 3x – 9 = x^2 – 9

(5 – y)(5 + y) = 25 +5y – 5y – y^2 = 25 – y^2

What do all of these have in common?

 ALL the Results are Difference of 2-Sqs:

Formula → ( A + B) ( A – B ) = A^2 – B^2