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The speed of light is 186,000 miles per second. How far can light travel in one minute? Write your answer in scientific notation.

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Review From Yesterday:

Like terms have the same exponent to the same degree.

When adding or subtracting, only like terms can be combined. When polynomials have more than one variable, the same rules apply. For example:

(xz + 5x²z – x) + (x+ 5z)

(x + z) + (zx+ z²x)

(a + 5ba) + (3ba+ a²)

(3xyz - xyz + zx) + (3zyx+ 1)

Like terms have the same exponent to the same degree.......But order does not matter!

Multiplying Polynomials:

Let's begin by multiplying a monomial by a monomial.

= 2x7

(2x3) • (yx4) =

If the bases (x) are the same, we add the

exponents 2x7yNow multiply a monomial by a binomial

Once more: x(7x2 + 4y) =

7x3 + 4x

7x3 + 4xyWhen multiplying polynomials, each term is

multiplied by every other term.

Now we look at multiplying a binomial by a binomial.

Method #1: The Box Method

Multiplying Binomials

(x + 4)(x + 2)

*Reminder: When multiplying, add the exponents if bases are alike

Binomials

Multiplying Binomials

=

=

Use the Box Method:

x2 -3x

+4x -12

Multiplying Binomials

=

However, the More Common Method for solving binomials is...

=

The Foil Method

F.O.I.L

F.O.I.L.

(x + 1) (x + 2) = x ( x + 2 ) + 1 ( x + 2 )

If we perform our distribution in this order,

First + Outer + Inner + Last

a useful pattern emerges.

(x + 1)(x + 2) = x (x + 2) + 1 (x + 2)

Distributing produces the sum of these four multiplications.

"F.O.I.L" for short.

x2 + 2x + x + 2

x2 + 3x + 2

Multiplying Binomials Mentally

(x + 2)(x + 1)

(x + 3)(x + 2)

(x + 4)(x + 3)

(x + 5)(x + 4)(x + 6)(x + 5)

x2 + x + 2x + 2

x2 + 2x + 3x + 6

x2 + 3x + 4x + 12

x2 + 4x + 5x + 20x2 + 5x + 6x + 30 x2 + 11x + 30

x2 + 9x + 20

x2 + 7x + 12

x2 + 5x + 6

x2 + 3x + 2

The middle term of the answer is the sum of the binomial's last terms and the last term in the answer is the product of the binomial's

last terms.

(x + a)(x + b) = x2 + x(a + b) + abThere are lots of patterns here, but this one

enables us to multiply binomials mentally.

Find the pattern

Positive and NegativeAll of the binomials we have multiplied so far have been sums of positive numbers. What happens if one of the terms is negative?

Example 1:

1. The last term will be negative, because a positive times a negative is negative.2. The middle term in this example will be positive, because 4 + (- 3) = 1.

Example 2:

(x + 4)(x - 3)

1. The last term will still be negative, because a positive times a negative is negative.2. But the middle term in this example will be negative, because (- 4) + 3 = - 1.

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

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

(x - 4)(x + 3)

Two NegativesWhat happens if the second term in both binomials is negative?

Example:

1. The last term will be positive, because a negative times a negative is positive.

2. The middle term will be negative, because a negative plus a negative is negative.

(x - 4)(x - 3)

(x - 4)(x - 3) = x2 -7x +12

Compare this result to what happens when both terms are positive:

(x + 4)(x + 3) = x2 +7x +12

Both signs the same:

last term positivemiddle term the same

Sign Summary

(x + 4)(x + 3)

Middle Term

Last Term

positive

positive

(x - 4)(x + 3)

negative

negative

(x + 4)(x - 3)

positive

negative

(x - 4)(x - 3)

negative

positive

Which term is larger doesn't matter when both signs are the same, but it does when the signs are different.

Remember, F.O.I.L can be used when multiplying a binomial by another binomial.

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