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### Warning on today’s homework:

Now please

CLOSE

YOUR LAPTOPS

and turn off and put away your cell phones.

Sample

Problems

Page Link

(Dr. Bruce Johnston)

Coming up:

- Today:
- Lecture on Section 5.4
- Next Class Session:
- HW 5.4 due at start of class
- Review Lecture on Sections 4.1 & 5.1-5.4
- Class Session After That:
- Practice Quiz 3 due at start of class
- Quiz 3 in class

Section 5.4Multiplying 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

= (4x2)(3x2) + (4x2)(-2x) + (4x2)(5)

(distributive property)

= 12x4 – 8x3 + 20x2

(multiply the monomials)

Multiply each of the following:

1) (3x2)(-2x)

= (3 • -2)(x2 • x)

= -6x3

2) (4x2)(3x2 – 2x + 5)

3) (2x – 4)(7x + 5)

= 2x(7x + 5) – 4(7x + 5)

(can also use “FOIL” on this) = 14x2 + 10x – 28x – 20

= 14x2 – 18x – 20

Multiply (3x + 4)2

Remember that a2 = a• a, so (3x + 4)2 = (3x + 4)(3x + 4).

Example

(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 (3x – 7y)(7x + 2y)

(3x – 7y)(7x + 2y)

= (3x)(7x + 2y) – 7y(7x + 2y)

= 21x2 + 6xy – 49xy + 14y2

= 21x2 – 43xy + 14y2

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

When multiplying 2 binomials, the distributive property can be easily remembered as the FOIL method.

F – product of First terms

O – product of Outside terms

I – product of Inside terms

L – product of Last terms

= y2 – 8y – 48

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

F O I L

y2 + 4y – 12y – 48

(y – 12)(y + 4) =

Example

L

F

F

O

I

L

2x(7x)

+ 2x(5)

– 4(7x)

– 4(5)

I

O

Multiply(2x – 4)(7x + 5)

(2x – 4)(7x + 5) =

= 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.

- 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.)

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.

Example

Techniques of multiplying polynomials are often useful when evaluating polynomial functions at polynomial values.

If f(x) = 2x2 + 3x – 4, find f(a + 3).

We replace the variable x with a + 3 in the polynomial function.

f(a + 3) = 2(a + 3)2 + 3(a + 3) – 4

= 2(a2 + 6a + 9) + 3a + 9 – 4

= 2a2 + 12a + 18+3a + 9 – 4

= 2a2 + 15a + 23

Problem # 6 has an error in the computer answer key for one version of the problem.

If you get your answer marked wrong and you see a “-0x” term in the computer’s answer, just click “similar problem” and do a new version.

Reminder:

Next class session:

- HW 5.4 due at start of class
- Review Lecture on Sections 4.1 & 5.1-5.4

If you take Practice Quiz 3 at least once before next class session, you’ll get a lot more out of that review session.

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