Factors and Product in Math

Introduction to factors in math:
Divisor, an integer which evenly divides a number without leaving a remainder
Factorization, the decomposition of an object into a product of other objects
Integer factorization, the process of breaking down a composite number into smaller non-trivial divisors
A coefficient, a multiplicative factor in an expression, usually a number
A von Neumann algebra with a trivial center
Factor (graph theory), a spanning sub graph
(Source: Wikipedia)

Example Problem for Factors and Product in Math

Factors and product in math example problem 1:

Find the factors of the given quadratic equation  x2 - 11x - 180

Solution:

Given quadratic equation is x2 - 11x - 180

First factorize the given equation, we get

(x2 - 11x - 180) = (x2 - 20x + 9x - 180)

Grouping the first two terms and second two terms, we get

= (x2 - 20x) + (9x - 180)

= x (x - 20) + 9 (x - 20)

= (x - 20) (x + 9)


The factors of the given quadratic equation is (x - 20) and (x + 9)

Answer:

The final answer is (x - 20) and (x + 9)

Factors and product in math example problem 2:

Find the factors of the given quadratic equation  x2 + 13x + 42

Solution:

Given quadratic equation is x2 + 13x + 42

First factorize the given equation, we get

(x2 + 13x + 42) = (x2 + 7x + 6x + 42)

Grouping the first two terms and second two terms, we get

= (x2 + 7x) + (6x + 42)

= x (x + 7) + 6 (x + 7)

= (x + 6) (x + 7)


The factors of the given quadratic equation is (x + 6) and (x + 7)

Answer:

The final answer is (x + 6) and (x + 7)

Factors and product in math example problem 3:

Find the product of the given two factors (x - 2) and (x + 3)

Solution:

Given two factors are (x - 2) and (x + 3)

Multiply the each term of the two factors together, we get

(x - 2) * (x + 3) = (x * (x + 3)) - (2 * (x + 3))

= x2 + 3x - 2x - 6

= x2 + x - 6

The product of the two factors are x2 + x - 6

Answer:

The final answer is x2 + x - 6

Factors and product in math example problem 4:

Find the product of the given two factors (x - 4) and (x + 6)

Solution:

Given two factors are (x - 4) and (x + 6)

Multiply the each term of the two factors together, we get

(x - 4) * (x + 6) = (x * (x + 6)) - (4 * (x + 6))

= x2 + 6x - 4x - 24

= x2 + 2x - 24

The product of the two factors are x2 + 2x - 24

Answer:

The final answer is x2 + 2x - 24

Practice Problems for Factors and Product in Math

Factors and product in math practice problem 1:

Find the factors of the given quadratic equation  x2 - 31x - 360

Answer:

The final answer is (x - 40) (x + 9)

Factors and product in math practice problem 2:

Find the product of the given two factors (x + 4) and (x + 3)

Answer:

The product of the two factors are x2 + 7x + 12

Determining Polynomial Functions

Introduction about polynomial:

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminate) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 - 4x + 7 is a polynomial, but x2 - 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number. (Source: Wikipedia)

Example Problems for Determining Polynomial Functions :

Determining polynomial functions - Example problem 1:

Simplify f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y)

Solution:

f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y) = 21x + 15y + 9x – 21y – 11x + 17y

= 21x + 9x - 11x + 15y – 21y + 17y

= 19x +11y

The answer for f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y) = 19x + 11y

Determining polynomial functions - Example problem 2:

Simplify f(x) = (15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21)

Solution:

(15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21)

= 15x^2 – 11x – 21 + 7x^2 – 16x – 19 + 7x^2 - 5x - 21

= 15x^2 + 7x^2 + 7x^2 – 11x – 16x - 5x – 21 – 19 - 21

= 29x^2– 36x – 61

The answer for f(x) = (15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21) = 29x^2– 36x – 61

Determining polynomial functions - Example problem 3:

Simplify f(x) = (14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21)

Solution:

(14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21)

= (14x3 + 8x^2 + 21x – 16) – 1(18x3 – 15x^2 – 7x + 17) – 1(8x^2 – 11x – 21)

= (14x3 + 8x^2 + 21x – 16)– 1(18x3) – 1 (–15x^2) – 1(–7x) – 1(17) - 1(8x^2) -1(-11x) -1(-21)

= 14x3 + 8x^2 + 21x – 16 – 18x3 + 15x^2 + 7x – 17 - 8x^2 + 11x + 21

= 14x3 – 18x3 + 8x^2 + 15x^2 -8x^2 + 21x + 7x + 11x – 16– 17 + 21

= -4x3 + 15x^2 + 39x –20

The answer for f(x) = (14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21) = -4x3 + 31x^2 + 39x –20

Practice Problems for Determining Polynomial Functions :

1. Simplify f(x)= (15x + 8y) + ( 7x – 15y ) – (8x – 9y)

The answer for f(x)= (15x + 8y) + ( 7x – 15y ) – (8x – 9y) = 14x + 2y

2. Simplify f(x)= (7x^2 – 9x – 8) + (3x^2 – 8x – 11) - (–6x^2 + 9x + 15)

The answer for f(x)=(7x^2 – 9x – 8) + (3x^2 – 8x – 11) - (–6x^2 + 9x + 15)= 16x^2– 26x – 34

Radius of a Circle Calculator

Introduction to radius of a circle calculator:

In mathematics, radius of a circle calculator is the calculator used to find the radius of the circle. In the radius of a circle calculator, we can find the radius value from any of the two values either by area or circumference. In the radius of a circle calculator, we can get the value of radius by just giving the value of circumference or area. The steps used in radius of a circle calculator are given below.

Radius of a Circle Calculator - Steps:

The diagram for the circle is given below:



Steps used in radius of a circle calculator are given below:

Step 1: Enter the value of circumference or area in the box given in the calculator.

Step 2: Then press "calculate" button.

Step 3: After some more second the process complete and shows the value of radius in the calculator.

Some of the examples based on this calculator is given below.

Radius of a Circle Calculator - Examples:

Example 1: Find the radius of the circle given that the area of circle is 153.86?

Solution:

Steps used in radius of a circle calculator  are given below:

Step 1: Enter the value of area as 153.86 in the box given in the calculator.

Step 2: Then press "calculate" button.

Step 3: After some more second the process complete and shows the value of radius as 7 in the calculator.I like to share this 6th grade math problems online with you all through my article.

Example 2: Find the radius of the circle given that the circumference of circle is 56.52?

Solution:

Steps used in radius of a circle calculator  are given below:

Step 1: Enter the value of circumference as 56.52 in the box given in the calculator.

Step 2: Then press "calculate" button.

Step 3: After some more second the process complete and shows the value of radius as 9 in the calculator.

Radius of a circle calculator – Practice problem:

Problem 1: Find the radius of the circle given that the area of circle is 379.94?

Answer is given below:

The output for that calculator is given below:



Problem 2:  Find the radius of the circle given that the circumference of circle is 81.64?

Answer is given below:

The output for that calculator is given below:

Solving Variable Expressions

Introduction to solving variable expressions:

In mathematics, an expression is a finite combination of symbols that are well-formed according to the rules applicable in the context at hand. Symbols can designate values (constants), variables, operations, relations, or can constitute punctuation or other syntactic entities. The use of expressions can range from simple arithmetic operations like 3+ 5 x ((-2)^7 – 3/2) . (Source: Wikipedia)

Types of Math Solving Variable Expressions:-

In solving variable expressions to study the algebra expressions in following types are used in algebra expressions.

Variable expressions Variable expressions using word

Variable expressions  

In variable expressions means to form a number and word in expression like as add, plus, greater, less than, increase, decrease etc.

For Example,

24 increased by x?

24 + x

Variable expressions using word

In variable expressions using word means the expression numbers and word are shows the sentence formation like as add, plus , greater ,less than ,decrease etc

For Example,

Haley earned 31 bonus points. Marisol earned b more bonus points than Haley. Choose the expressions that show how many bonus points Marisol earned.

31+b

Example Problems for Solving Variable Expressions:-

Problem 1:-

Solving variable expressions for 370 added to v.

Solution:-

Adding the variable v to number 370 and form an expression as V+370

Answer:- V+370 

Problem 2:-

Solving variable expression for 28 minus w

Solution:-

Subtracting the variable w to number 28 and form an expression as 28-w. I like to share this Algebra 2 problem solver with you all through my article.

Answer: 28 –w

Problem 3:-

Solving variable expression for 703 increased by z

Solution:-

Increased the variable z to number 703 and form an expression as 703+z

Answer: 703+z

Problem 4-

Solving variable expression for 45 decreased by p

Solution:-

Decreased the variable p to number 45 and form an expression as 45-p

Answer: 45-p

Problem 5-

Talia earned 64 bonus points. Jones earned d more bonus points than Talia. Choose the expression that shows how many bonus points Jones earned.

Solution:-

Adding the jones variable d to talia earned 64 bonus points and form an expression as 64+d

Answer: 4+d

Vector Component Calculator

Introduction to vector component calculator:

The collection of ordered components are called as vector and the types of components are x component and y component. The calculator is a device that gives the output of all math operations. The vector component is also determined by calculator. Both vector components are representing the direction by sign. Now we are going to see about vectors component calculator.

Explanation for Vector Component Calculator

Vector component:

In math, the vector component is used in algebra. The x component is representing the horizontal component that is x-axis value and the y component is representing the vertical component that is y-axis value. If two vectors are present means find out the components by addition operation.

Vector component calculator:

The vector calculator is calculating the components by addition method. The x component of first vector is added with x component of second vector. Similarly the y component of first vector and second vector is added.I like to share this math homework help answers with you all through my article.

The graph representation is used in vector calculator. The calculator contains graph with in it. We can see the calculator operations stepwise.

More about Vector Component Calculator

Following steps are operations of vector calculator:

Step 1: First plotting the first vector in graph.



The first vector is plotted as above and this vector is represented with x components 5 and y components 4.

Step 2: Plotting the second vector in graph.



The second vector is plotted as above diagram.The second vector is represented with x component 6 and y component -4.

Step 3: The addition of two vectors.



The given two vectors are added by calculator. The result of vector addition in calculator is shown above.

The calculator represents the x component value is 11 that is 5 + 6 and the y component is 0 that is 4 - 4. Another application of vector calculator is magnitude and direction determination.

Array Math Definitions

Introduction to array

Let us discuss about the array math definition. The definition of array is number or objects arranged in rows and columns. The array is the main tools in the mathematics. The basic operation of the array is multiplication and division. The array is declaring the square bracket. The example of the array is [8]. Next we see the definition of an array.

Definition of the Array in Math

The definition of the diagonal array is square matrix.  A diagonal matrix having the element only in the diagonal position. The remaining position elements are zero.

`[[9,0,0],[0,5,0],[0,0,7]]`

The array of number is 3 rows and 3 columns. That is defining the B [3] [3].

Definition of row matrix

The definition of row matrix is called as the only one row. It is represents the 1 x n. the n is declare the number of columns.

Examples of row matrix: [4   5   9]

Math definition of column matrix

The column matrix is defined as the matrix contains only one column. It is declaring the up to down format. It is represents the n x 1. The n is called as the number of rows.

Examples of column matrix:

`[[8],[11],[14]]`

Math Definition of even Array

The definition of the even array is the number of rows and column is a even number. the even number is normally divisible by two. The example of the even number is 2, 4, 6 etc. the example of the even array is,

`[[5,8],[7,6]]`

This array is the 2 x 2 array.

The definition of the 2 x 2 array is declare only 2 rows and 2 column should be followed.

2 x 2 = 2 + 2

= 4.    

Sample E Learning

Introduction to sample e learning:-

Sample e learning is important for students. Student’s does learning the sample e learning and also solve the e learning problems. Here e means `e^x` . In math exponential function means ex, where e is the significance of ex the same value again consequent.
For example,
`17e^x` this is a way to write an exponential function.
e = 2.718 is the value of e.

Basic Properties of Sample E Learning:-

In the following basic properties of  sample e learning:

`e^x e^y = e^(x+y)`
`(e^x)^p = e^(px)`
`(de^x)/(dx) = e^x`
`(de^ax)/(dx) = ae^(ax)`
`(d^n e^ax)/(dx^n) = a^n e^(ax)`
`e^x/ e^y = e^(x-y)`
`root(p)(e^x) = e^(x/p)`
`inte^x dx = e^x `

Example Problems to Sample E Learning:-

Problem 1:-

Solve the exponential function equation `e^x = 47`

Solution:-

Here the natural log is the inverses of exponential function, so use ln to get fast solve this problem.

ln `e^x` = ln 47

x = ln 47 (take natural log of 47)

= 3.850

So the answer is 3.850


Problem 2:-

Solving add the exponential equation `e^(18x) +e^(11x)`

Solution:

Given: `e^(18x) +e^(11x)`

We know the property `e^x e^y = e^(x+y)`

Take the common term e.

= `e^(18x+11x)`

= `e^(29x)`

Adding the both values and get 29x.

Finally we get an answer as `e^(29x)`


Problem 3:-

Solving whether the point (0, 1) lies on the graph of the function y = 18(4)x.

Solution:-

Substitute x = 0 in the function y = 18(4)x.

We know the property `e^x`

y = 18(4)0

= 18(1)

= 18

The y–coordinate of the point is 1, which does not match with the obtained value y = 18.

So, the graph of the function y = 18(4)x does not contain the point (0, 1).