Transcript Please open your laptops, log in to the MyMathLab course

Please open your laptops, log in to
the MyMathLab course web site,
and open Quiz 5.2.
You may use the formula sheet on this quiz – please don’t write on this sheet,
and remember to hand it back in with your quiz answer sheet.
Please
CLOSE
YOUR LAPTOPS,
and turn off and put away your
cell phones,
and get out your notetaking materials.
Sections 5.3/5.4
Multiplying Polynomials
Multiplying Polynomials
• Multiplying polynomials
• If all of the polynomials are monomials, use
the associative and commutative properties,
along with properties of exponents.
• If any of the polynomials have more than one
term, use the distributive property before the
associative and commutative properties. Then
combine like terms.
Example
Multiply each of the following:
1) (3x2)(-2x) = (3 • -2)(x2 • x) = -6x3
2) (4x2)(3x2 – 2x + 5)
= (4x2)(3x2) + (4x2)(-2x) + (4x2)(5)
= 12x4 – 8x3 + 20x2
3) (2x – 4)(7x + 5)
(can also use “FOIL” on this)
(distributive property)
(multiply the monomials)
= 2x(7x + 5) – 4(7x + 5)
= 14x2 + 10x – 28x – 20
= 14x2 – 18x – 20
Example
Multiply (3x + 4)2
Remember that a2 = a • a, so (3x + 4)2 = (3x + 4)(3x + 4).
(3x + 4)2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4)
=
9x2 + 12x + 12x + 16
=
9x2 + 24x + 16
EXTREMELY IMPORTANT NOTE:
(3x + 4)2 is NOT simply (3x)2 + 42 !!!
Example
Multiply (a + 2)(a3 – 3a2 + 7).
(a + 2)(a3 – 3a2 + 7) = a(a3 – 3a2 + 7) + 2(a3 – 3a2 + 7)
=
a4 – 3a3 + 7a + 2a3 – 6a2 + 14
=
a4 – a3 – 6a2 + 7a + 14
Example
Multiply (5x – 2z)2
(5x – 2z)2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – 2z(5x – 2z)
= 25x2 – 10xz – 10xz + 4z2
= 25x2 – 20xz + 4z2
REMINDER:
(5x -2z)2 is NOT simply (5x)2 – (2z)2 !!!
Example
Multiply (2x2 + x – 1)(x2 + 3x + 4)
(2x2 + x – 1)(x2 + 3x + 4)
= (2x2)(x2 + 3x + 4) + x(x2 + 3x + 4) – 1(x2 + 3x + 4)
=
2x4 + 6x3 + 8x2 + x3 + 3x2 + 4x – x2 – 3x – 4
=
2x4 + 7x3 + 10x2 + x – 4
Special Products
• Some types of polynomial products can be carried
out more efficiently using techniques that apply only
to specific situations such as two binomials or
squaring a binomial.
• These products can also be calculated using the basic
laws of exponents and the distributive property, but
these shortcuts may save you some time if you can
learn to recognize the situations to which they apply.
• The shortcuts are based on the laws of exponents
and the distributive property, as you will see as we
go through the sample problems.
When multiplying 2 binomials, the distributive property
can be easily remembered as the FOIL method.
This is just a memory device that may be useful to keep you from
forgetting any of the four parts of the product.
FOIL only applies to a binomial (two-term polynomial) multiplied
by another binomial, not to any other types of products
involving monomials, trinomials, etc.
F – product of First terms
O – product of Outside terms
I – product of Inside terms
L – product of Last terms
Example
Multiply (y – 12)(y + 4)
(y – 12)(y + 4)
Product of First terms is y2
(y – 12)(y + 4)
Product of Outside terms is 4y
(y – 12)(y + 4)
Product of Inside terms is -12y
(y – 12)(y + 4)
Product of Last terms is -48
(y – 12)(y + 4) =
F O
I
L
y2 + 4y – 12y – 48
= y2 – 8y – 48
Example
Multiply (2x – 4)(7x + 5)
L
F
F
O
I
L
(2x – 4)(7x + 5) = 2x(7x) + 2x(5) – 4(7x) – 4(5)
I
O
= 14x2 + 10x – 28x – 20
= 14x2 – 18x – 20
In the process of using the FOIL method on products of
certain types of binomials, we see specific patterns that lead
to special products such as the following:
• Squaring a Binomial
• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
(These might be just as easy to do by the usual FOIL method rather than by
memorizing the formulas.)
• Multiplying the Sum and Difference of Two
Terms
• (a + b)(a – b) = a2 – b2
(This formula can be quite useful and save you some time.)
NOTE: These three formulas are on your formula sheet.
Problem from today’s homework:
• Although you will arrive at the same results
for the special products by using the
distributive property, memorizing these
products (especially the last one) can save
you some time in multiplying polynomials.
• Multiplying 3 or more polynomials together
might require you to use more than one
technique. Multiply the polynomials two at
a time.
Problem from today’s homework:
How would you attack this problem?
(x –
3
5)
How about this problem?
(2x –
2
3)(4x
+ 9)(2x + 3)
REMINDER:
The assignment on today’s material (HW 5.3/4) is
due at the start of the next class session.
Lab hours in 203:
Mondays through Thursdays
8:00 a.m. to 6:30 p.m.
Please remember to sign in on the Math 110 clipboard
by the front door of the lab
You may now OPEN
your LAPTOPS
and begin working on the
homework assignment.
We expect all students to stay in the classroom
to work on your homework till the end of the 55minute class period. If you have already finished
the homework assignment for today’s section,
you should work ahead on the next one or work
on the next practice quiz/test.