Multiplying Linear Functions

Multiply the following two linear functions:

(x+3)(2x+5)

Answer :




Explanation: The standard format of a Linear Function is as follows: f(x) = ax + b

It is important to keep in mind the following regarding Linear Functions:

1. The term “Linear” is derived from the word “Line”. Therefore a Linear Function is a function or equation of a straight line.

2. The degree of this function is always 1 (a degree means the highest power or exponent The variable in the function is raised to.

Therefore by looking at the two sets of functions presented in this problem we can identify (x+3) as one linear function and also (2x+5) as another linear function and the requirement of this problem is to multiply the two and end up with a resulting quadratic equation.

In order to perform the multiplication we use a method called FOIL which stands for:

First – Outer – Inner – Last

Lets now identify those terms in the above linear functions:

First terms : (x) in the first function (x+3) and the (2x) in the (2x+5) function

Outer terms : (x) in the first function (x+3) and the (5) in the (2x+5) function

Inner terms : (3) in the first function (x+3) and the (2x) in the (2x+5) function

Last terms : (3) in the first function (x+3) and the (5) in the (2x+5) function

Now let us implement the above multiplication:






Notice that the middle like terms of 5x and 6x are combined and thus results in 11x as the middle term for the resulting quadratic function.

© Copyright 2012 Najwa S. Hirn. All rights reserved.

Exponents - Multiplication..

Multiply the following and explain your answer:



Answer:



Explanation:

In Algebra, the term “Exponent” is also referred to as a “Power”. It is the small number that appears on the upper right hand corner of either a Variable or a Constant.

A Constant is an unchanged value, which normally is a number.
A Variable is a changed value and is usually referred to by an alphabet.

The Constant or the Variable are termed “The Base”.

Each one of the above expressions is referred to as a “Term”

Therefore

is a Term

and

is another Term.


The number “5” in each term is the Constant and the “a” is the Variable.

When multiplying terms it is important to be know the following:

1. The rule of multiplying exponents.
2. The signs of the terms

The rule for multiplying exponent states that when multiplying terms that have the same variables, the numbers are multiplied normally while the exponents on the variables are added.

The signs of each term must be multiplied to reach the end results.

In the above problem, we now know that we must multiply the number “5” in the first term by the number “5” in the second term. We also know that we must add the exponents that appear over the “a”. We should realize that both of those terms are positive since there is no other sign that appears before each term. That tells us that we are multiplying two positive numbers and positive variables which gives us a result of a positive answer.

Lets solve the problem (please remember that when terms are enclosed by a paranthesis then that means that you are multiplying the terms and normally you would not need to show the positive signs - they are shown below for explanantion purposes):









© Copyright 2009. Najwa S. Hirn. All rights reserved.

Slope Of A Line - College Level

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Find the slope of the line going through the following points:

(3, 6) and (-2, -4)

The term “Slope” usually refers to an incline of a straight line. In order to visualize a straight line that is drawn at an incline, it is important to note that this line will travel a certain horizontal distance while rising up or down a certain vertical distance. The horizontal distance is called the “run” and the vertical distance is called the “rise”. The ratio between the two is called the “slope” which also represents the steepness of that line.

The formula used for calculating the slope of a straight line is as follows:

Where m represents the slope going through two points that are indicated as:

In order to calculate the slope we use the above formula as follows:

Therefore, the slope of the line = 2

© Copyright 2008.Najwa S. Hirn. All rights reserved.

Solving for a Variable - Middle school math

Solve for t in the following equation:

3t + b = 5

When a problem asks to solve for a particular variable, it means to isolate that variable on one side of the equation (equal sign) and isolate the other terms on the other side of the equal sign.

In order to isolate the t term in the above problem, we must eliminate b from the left side of the equation. This can be accomplished if b is subtracted from both sides of the equal sign as follows:

3t + b – b = 5 – b

It is important to remember that whatever is done to the right side of the equation must be done to the left side also.

At this point b is eliminated from the right side of the equal sign and the resulting problem is:

3t = 5 – b

It is necessary at this time to eliminate the number 3 from the left side of the equation. There are two ways that may be used to accomplish this. Either multiply both side of the equation by (1/3) or divide both side of the equation by 3.

Lets divide each side by 3 as follows:

(3t) / 3 = (5 - b) / 3

The 3’s on the left side of the equation cancel out and the problem is finalized as follows:

t = (5 - b) / 3

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Number Sense - Elementary School Math

There were 35 pizzas at Andrew’s birthday party and 16 of them had only pepperoni topping while the rest had a mushroom topping. How many pizza had the mushroom topping?

a. 12
b. 19
c. 21
d. 8

Answer:  (b)  19 pizzass had only a mushroom topping.

The student needs to recognize number sense and know how to add, subtract, multiply and divide mathematical expressions.

In order to solve this problem, it is important to realize that the total number of pizza’s were 35 and since 16 had only pepperoni that means that the student must subtract 16 from the total of 35 to obtain the number of pizza’s with the mushroom topping.

35 – 16 = 19 pizzas with a mushroom topping.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Fractional Expressions - College Level Math

Multiply the following fractional expression and simplify:

Answer:

In order to solve this problem, we first must factor the numerator and denominator of each fraction. Once they are factored the problem will look like the following:

Now it is time to multiply the numerators and multiply the denominators as follows:

It is obvious from the above that several factors can be cancelled. It is noted that the(a-1) factors can be cancelled from both numerator and denominator and one (a +4) can be cancelled from the numerator. The final simplified answer will look as follows:

© Copyrightreserved 2008.Najwa S. Hirn. All rights

Volume Of A Cube - High School Math

One face of a cube has an area of 64 cm square – What is the volume of this cube?

Answer : 512 cm cubed

In order to calculate the volume of a cube, the length, height and width of the cube must be known since the formula for finding the volume is:
V = l w h
The problem does not provide information about the length, width or height; therefore they must be calculated using the information we were given about the area.

In this problem, we are given the area of one face. We know that the formula for calculating an area for any two dimensional figure is:
A = l w
Since each face of a cube is constructed from a square, then we know that the length and width of this square must be the same and by multiplying the two, the area can be found.

Lets apply what we have:
64 cm square = l w
Since we know that the length and width must be equal for a square, we can take the square root of the area in order to calculate the length of width:

square root of 64 = 8

Therefore, the length and width of each face is 8 cm. Therefore, the height of this cube will also be
8 cm. The volume can now be calculated as follows:

V = (8cm)(8cm)(8cm) = 512 cm cubed.

©Copyright 2008.Najwa S. Hirn. All rights reserved.