Week of September 4, 2018

Astronomy

Newton's Laws of Motion

The laws that govern motion eluded scientists, philosophers and other great thinkers until the 17th century. Then, in the 1680s, Isaac Newton proposed three laws that explained how inertia, acceleration and reaction influence the motion of objects. Along with Newton’s law of gravitation, these laws formed the basis of classical physics.

The Law of Inertia

Newton's first law of motion, also known as the law of inertia, states that objects neither move nor cease to move on their own. An object only changes its state of motion when acted on by an outside force. A ball at rest, for example, will remain at rest until you push it. It will then roll until friction from the ground and the air brings it to a halt.

The Law of Acceleration

Newton's second law explains how external forces affect the velocity of an object in motion. It states that acceleration of an object is directly proportional to the force that causes it, and inversely proportional to the object's mass. In practical terms, this means that it takes more force to move a heavy object than a light one.

Consider a horse and cart. The amount of force the horse can apply determines the cart's speed. The horse could move faster with a smaller, lighter cart in tow, but its maximum speed is limited by the weight of a heavier cart.

In physics, deceleration counts as acceleration. Thus, a force acting in the opposite direction of a moving object causes an acceleration in that direction. For example, if a horse is pulling a cart uphill, gravity pulls the cart downward as the horse pulls upward. In other words, the force of gravity causes a negative acceleration in the horse's direction of motion.

The Law of Reaction

Newton's third law states that for every action in nature, there is an equal and opposite reaction. This law is demonstrated by the act of walking or running. As your feet exert force down and backward, you are propelled forward and upward. This is known as "ground reaction force."

This force is also observable in the motion of a gondola. As the driver presses his punting pole against the ground beneath the water's surface, he creates a mechanical system that propels the boat forward along the water's surface with a force equal to that which he applied to the ground.

https://sciencing.com/newtons-laws-motion-7258289.html

FOUNDATIONS OF ALGEBRA

Multiplying Polynomials

Multiplying polynomials involves applying the rules of exponents and the distributive property to simplify the product. This multiplication can also be illustrated with an area model and can be useful in modeling real world situations. Understanding polynomial products is an important step in factoring and solving algebraic equations.

The Product of a Monomial and a Polynomial

The distributive property can be used to multiply a polynomial by a monomial. Just remember that the monomial must be multiplied by each term in the polynomial. Consider the expression 2x(2x² + 5x + 10).

This expression can be modeled with a sketch like the one below. This model is called an area model because the rectangular pieces represent the area created by the multiplication of the monomial and the polynomial.

	2x2	5x	10
2x	4x3	10x2	20x

We can see that the product of the width, 2x, and the length, 2x² + 5x + 10, is the area of the entire shaded region. The area can be split into three smaller pieces. Each of those pieces has a width of 2x and a length represented by one of the terms of the polynomial.

Area models are a helpful way to visualize a multiplication problem. But we can also find the product of two polynomials algebraically, by applying the distributive property. Remember that the distributive property says that multiplying a sum by a number is the same as multiplying each addend by the number and then adding: a(b + c) = ab + ac. It doesn't matter how many terms there are: a(b + c + d) = ab + ac + ad.

Let's try one:

Example
Problem	5x3(4x2 + 3x + 7)
			Distribute the monomial to each term of the polynomial
			Add the products
Answer

Product of Two Binomials

Now let's explore multiplying two binomials. Once again, we can draw an area model to help us make sense of the process. We'll use each binomial as one of the dimensions of a rectangle, and their product as the area.

The model below shows (x + 4)(2x + 2):

	x	1	1	1	1
x	x2	x	x	x	x
x	x2	x	x	x	x
1	x	1	1	1	1
1	x	1	1	1	1

Each binomial is expanded into individual variables and numbers, x + 4 along the top of the model and 2x + 2 along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, which is also the final product of multiplying the binomials. If we combine all the like terms, we can write the product, or area, as 2x² + 10x + 8.

We can also use algebra to determine the product of two binomials. Just multiply each term in one binomial by all the terms in the other binomial as shown below:

Example
Problem	(x + 4)(2x + 2)
	x(2x + 2) + 4(2x + 2)		Multiply each term in one binomial by each term in the other binomial
	2x2 + 2x + 8x + 8		Rewrite to group like terms together
	2x2 + 10x + 8		Combine like terms
Answer	2x2 + 10x + 8

Look back at the rectangle and see where each piece of 2x² + 2x + 8x + 8 comes from. Can you see where we multiply x by 2x + 2, and where we get 2x² from x(2x)?

Because multiplication is commutative, the terms can be multiplied in either order. The expression (2x + 2)(x  + 4) has the same product as (x  + 4)(2x + 2), both having a product of 2x² + 10x + 8. (Work it out and see.) The order in which we multiply binomials does not matter. What matters is that we multiply each term in one binomial by each term in the other binomial.

The last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.

http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U08_L2_T3_text_container.html