Category: Algebra
"Published in Newark, California, USA"
Find the equation of a polynomial if the roots are 3, 1, and -√2.
Solution:
If
one of the root of the equation is given which is -√2 , then we need
to get its conjugate because we want to eliminate the radical sign
in the given equation. The conjugate of -√2 is √2 . Hence, the
equation or a factor from the product of a root and its conjugate is
Therefore, the equation of a polynomial is
This website will show the principles of solving Math problems in Arithmetic, Algebra, Plane Geometry, Solid Geometry, Analytic Geometry, Trigonometry, Differential Calculus, Integral Calculus, Statistics, Differential Equations, Physics, Mechanics, Strength of Materials, and Chemical Engineering Math that we are using anywhere in everyday life. This website is also about the derivation of common formulas and equations. (Founded on September 28, 2012 in Newark, California, USA)
Showing posts with label Algebra. Show all posts
Showing posts with label Algebra. Show all posts
Saturday, December 6, 2014
Friday, December 5, 2014
Theory of Equations, 7
Category: Algebra
"Published in Newark, California, USA"
Find the remaining roots of the equation
if 1 - i√2 is a root.
Solution:
If one of the root of the equation is given which is 1 - i√2 , then we need to get its conjugate because we want to eliminate the imaginary number and the radical sign in the given equation. The conjugate of 1 - i√2 is 1 + i√2 . Hence, the equation or a factor from the product of a root and its conjugate is
In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have
Since there's no remainder in the division, then the other factor of the given equation is x² + 2x - 3.
Let's factor the other factor of the given equation as follows
If you equate each factor to zero, then the values of x are -3 and 1.
Therefore, the other roots of the given equation are -3 and 1.
"Published in Newark, California, USA"
Find the remaining roots of the equation
if 1 - i√2 is a root.
Solution:
If one of the root of the equation is given which is 1 - i√2 , then we need to get its conjugate because we want to eliminate the imaginary number and the radical sign in the given equation. The conjugate of 1 - i√2 is 1 + i√2 . Hence, the equation or a factor from the product of a root and its conjugate is
In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have
Since there's no remainder in the division, then the other factor of the given equation is x² + 2x - 3.
Let's factor the other factor of the given equation as follows
If you equate each factor to zero, then the values of x are -3 and 1.
Therefore, the other roots of the given equation are -3 and 1.
Thursday, December 4, 2014
Theory of Equations, 6
Category: Algebra
"Published in Newark, California, USA"
Find the remaining roots of the equation
if 2 + √5 is a root.
Solution:
If one of the root of the equation is given which is 2 + √5 , then we need to get its conjugate because we want to eliminate the radical sign in the given equation. The conjugate of 2 + √5 is 2 - √5. Hence, the equation or a factor from the product of a root and its conjugate is
In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have
Since there's no remainder in the division, then the other factor of the given equation is 2x + 3.
Therefore, the other root of the given equation is .
"Published in Newark, California, USA"
Find the remaining roots of the equation
if 2 + √5 is a root.
Solution:
If one of the root of the equation is given which is 2 + √5 , then we need to get its conjugate because we want to eliminate the radical sign in the given equation. The conjugate of 2 + √5 is 2 - √5. Hence, the equation or a factor from the product of a root and its conjugate is
In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have
Since there's no remainder in the division, then the other factor of the given equation is 2x + 3.
Therefore, the other root of the given equation is .
Wednesday, December 3, 2014
Theory of Equations, 5
Category: Algebra
"Published in Newark, California, USA"
Find the remaining roots of the equation
if -√2 is a root.
Solution:
If one of the root of the equation is given which is -√2, then we need to get its conjugate because we want to eliminate the radical sign in the given equation. The conjugate of -√2 is √2. Hence, the equation or a factor from the product of a root and its conjugate is
In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have
Since there's no remainder in the division, then the other factor of the given equation is x² - 2x + 4.
By using the completing the square method, we can solve for the other roots of the given equation as follows
Therefore, the other roots of the given equation are and .
"Published in Newark, California, USA"
Find the remaining roots of the equation
if -√2 is a root.
Solution:
If one of the root of the equation is given which is -√2, then we need to get its conjugate because we want to eliminate the radical sign in the given equation. The conjugate of -√2 is √2. Hence, the equation or a factor from the product of a root and its conjugate is
In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have
Since there's no remainder in the division, then the other factor of the given equation is x² - 2x + 4.
By using the completing the square method, we can solve for the other roots of the given equation as follows
Therefore, the other roots of the given equation are and .
Tuesday, December 2, 2014
Theory of Equations, 4
Category: Algebra
"Published in Newark, California, USA"
Find the remaining roots of the equation
if 5 + i is a root.
Solution:
If one of the root of the equation is given which is 5 + i, then we need to get its conjugate because we want to eliminate the imaginary number in the given equation. The conjugate of 5 + i is 5 - i. Hence, the equation or a factor from the product of a root and its conjugate is
In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have
Since there's no remainder in the division, then the other factor of the given equation is x + 2.
Therefore, the other root of the given equation is - 2.
"Published in Newark, California, USA"
Find the remaining roots of the equation
if 5 + i is a root.
Solution:
If one of the root of the equation is given which is 5 + i, then we need to get its conjugate because we want to eliminate the imaginary number in the given equation. The conjugate of 5 + i is 5 - i. Hence, the equation or a factor from the product of a root and its conjugate is
In order to get the other factor for the given equation, let's divide the given equation with the above equation, we have
Since there's no remainder in the division, then the other factor of the given equation is x + 2.
Therefore, the other root of the given equation is - 2.
Monday, December 1, 2014
Square, Rectangle, and Parallelogram Problems, 12
Category: Algebra, Plane Geometry
"Published in Newark, California, USA"
In 1964, Mr. Tambasen bought a rectangular lot in Bacolod City for ₱ 18,000 and spent ₱ 1,000 for building a wall around it. If the cost per square meter of the lot is ₱ 30 and the cost per linear meter of the wall is ₱ 10, find the dimensions of the lot.
Solution:
To illustrate the problem, it is better to draw the figure as follows
The first working equation which is the cost of a rectangular lot is
The second working equation which is the total cost of building a wall around the rectangular lot is
Substitute the value of y to the first working equation, we have
By using the completing the square method, the value of x which is the length of a rectangular lot is
If you choose the positive sign, the value of x is
and the value of y which is the width of a rectangular lot is
If you choose the negative sign, the value of x is
and the value of y which is the width of a rectangular lot is
Therefore, the dimensions of a rectangular lot are 30 m by 20 m.
"Published in Newark, California, USA"
In 1964, Mr. Tambasen bought a rectangular lot in Bacolod City for ₱ 18,000 and spent ₱ 1,000 for building a wall around it. If the cost per square meter of the lot is ₱ 30 and the cost per linear meter of the wall is ₱ 10, find the dimensions of the lot.
Solution:
To illustrate the problem, it is better to draw the figure as follows
Photo by Math Principles in Everyday Life |
Let x = be the length of a rectangular lot
y = be the width of a rectangular lot
The area of a rectangular lot is .
The perimeter of a rectangular lot is .
The first working equation which is the cost of a rectangular lot is
The second working equation which is the total cost of building a wall around the rectangular lot is
Substitute the value of y to the first working equation, we have
By using the completing the square method, the value of x which is the length of a rectangular lot is
If you choose the positive sign, the value of x is
and the value of y which is the width of a rectangular lot is
If you choose the negative sign, the value of x is
and the value of y which is the width of a rectangular lot is
Therefore, the dimensions of a rectangular lot are 30 m by 20 m.
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