Basic Algebraic Concepts
COURSE OUTCOMES
Create visualizations and solutions for some problems involving basic number theory concepts, basic algebraic concepts, equations, inequalities, systems of linear equations, and matrices with the use of computer applications.
Basic Algebraic Concepts
LEARNING OUTCOMES
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Recognize the basic concepts involving
algebra, equations, inequalities, systems of linear
equations, and matrices -
Develop visualizations and solutions for
some problems involving equations, inequalities,
systems of linear equations, and matrices
ALGEBRA
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lgebra is a branch of mathematics that involves using letters and symbols to represent numbers and quantities. It is a powerful tool that enables us to solve complex problems that involve unknown values.
TERM
refers to a group of numbers, variables, or a combination of both that are separated by mathematical operators.
For example, in the expression 3x + 2y - 5, there are three terms: 3x, 2y, and -5.
VARIABLE
is a symbol or letter that represents a quantity that can change its value. In algebra, variables are often represented by letters such as x, y, or z.
in the equation 3x + 5 = 11, x is a variable.
CONSTANT
is a number that does not change its value. In algebraic expressions or equations, constants are represented by fixed values.
in the expression 2x + 5, 5 is a constant.
EQUATION
is a mathematical statement that shows that two expressions are equal. It contains an equals sign (=) that separates the left-hand side from the right-hand side.
2x + 3 = 9 is an equation.
NUMERICAL COEFFICIENT
is the numerical value that multiplies a variable in that term.
in the term 4x, the numerical coefficient is 4.
EXPRESSION
is a combination of numbers, variables, and mathematical operations. It does not contain an equals sign and cannot be solved on its own.
3x + 2y - 5 is an expression.
POLYNOMIALS
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is a mathematical expression consisting of terms with variables raised to powers and coefficients, which are separated by addition or subtraction signs. They are important in mathematics and can be used to model complex relationships between variables.
DEGREE OF POLYNOMIALS
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is the highest exponent of the variable in that polynomial. It is the value of the power of the variable that has the highest value.
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4y^5 - 3y^3 + 2y^2 + 7y - 1. In this case, the degree of the polynomial is 5, because the highest exponent of the variable y is 5.
TYPES OF POLYNOMIALS
CONSTANT POLYNOMIAL
is a polynomial of degree zero that contains only a single constant term.
3, -5, or 2/3 are constant polynomials.
CUBIC POLYNOMIAL
is a polynomial of degree 3, which means it contains one variable term raised to the third power.
2x^3 + 3x^2 - x + 4, -4y^3 + 5y^2 + 2y - 3, or 6z^3 - 4z^2 + z - 2 are cubic polynomials.
LINEAR POLYNOMIAL
is a polynomial of degree 1, which means it contains one variable term raised to the first power.
2x + 3, -5y + 1, or 4z - 2 are linear polynomials.
HOMOGENEOUS POLYNOMIAL
is a polynomial in which all the terms have the same degree.
x^2 + 2xy + y^2 is a homogeneous polynomial of degree 2, since all its terms are of degree 2.
QUADRATIC POLYNOMIAL
is a polynomial of degree 2, which means it contains one variable term raised to the second power.
3x^2 + 2x + 1, -2y^2 + 3y - 1, or 4z^2 - 2z + 7 are quadratic polynomials.
KINDS OF POLYNOMIALS
MONOMIAL
is a polynomial with only one term. The term may be a constant or a product of a constant and one or more variables raised to some powers.
5x, 3y^2, and -2z^3 are all monomials.
MULTIVARIATE POLYNOMIAL
is a polynomial with two or more variables. The terms may be constants or products of constants and variables raised to some powers.
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2x^2y + 3xy^2 - 4x + 7y, -5x^2y^2 + 2xy + 7z^3, and 4x^3y^2z - 6x^2yz^2 + 3t^2 are all multivariate polynomials.
BINOMIAL
is a polynomial with two terms. The terms may be constants or products of a constant and one or more variables raised to some powers.
3x + 2, -4y^2 + 7z, and 5t - t^2 are all binomials.
TRINOMIAL
is a polynomial with three terms. The terms may be constants or products of a constant and one or more variables raised to some powers.
2x^2 + 3x - 4, -5y^2 + 2y + 7z, and 4t^3 - 6t^2 + t are all trinomials.
OPERATIONS OF POLYNOMIALS
ADDITION
simply add the like terms.
(2x^2 + 3x - 4) + (4x^2 - 2x + 1)
= 6x^2 + x - 3
DIVISION
use long division or synthetic division.
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(3x^2 + 2x - 1) ÷ (x - 1) = 3x + 5 + 4/(x - 1).
SUBTRACTION
distribute the negative sign to each term in the second polynomial and then add the like terms.
(3x^2 - 2x + 1) - (2x^2 + 4x - 3) = x^2 - 6x + 4
FACTORING
find the common factors, use the difference of squares formula, or use other factoring methods.
x^2 - 4 can be factored as (x + 2)(x - 2), and 3x^2 + 4x - 4 can be factored as (3x - 2)(x + 2).
MULTIPLICATION
use the distributive property and then combine like terms.
(2x - 3)(3x + 4) = 6x^2 + 5x - 12
FINDING ROOTS
set the polynomial equal to zero and solve for the variable.
the roots of 2x^2 + 3x - 2 = 0 are x = -1/2 and x = 2/1.
SYSTEMS OF LINEAR EQUATIONS
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is a set of two or
more linear equations that involve the same
variables.
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2x + 3y = 7
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4x - 5y = -2
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This is a system of two linear equations in two variables, "x" and "y". To solve this system, we need to find the values of "x" and "y" that make both equations true at the same time.
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Three (3) types of Linear Equations are:
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Exactly 1 Solution
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Infinitely Many Solutions
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No Solution
METHODS
SUBSTITUTION METHOD
involves solving one equation for one variable and then substituting that expression into the other equation. The resulting equation will have only one variable and can be solved for its value.
For example, consider the system of equations: 2x + 3y = 7 x - y = 1
Using substitution, we can solve for x in the second equation: x = y + 1
Substituting this expression for x in the first equation, we get: 2(y + 1) + 3y = 7
Simplifying and solving for y, we get: y = 1
Substituting y = 1 into x = y + 1, we get x = 2.
Therefore, the solution to the system of equations is (x, y) = (2, 1).
ELIMINATION METHOD
involves adding or subtracting the equations in a system to eliminate one of the variables.
For example, consider the system of equations: 2x + 3y = 7 4x - 6y = 10
To eliminate y, we can multiply the first equation by 2 and the second equation by 3, giving us: 4x + 6y = 14 12x - 18y = 30
Now, subtracting the first equation from the second equation, we get: 8x - 24y = 16
Dividing both sides by 8, we get: x - 3y = 2/3
Substituting this expression for x into the first equation, we get: 2(2/3 + 3y) + 3y = 7
Simplifying and solving for y, we get: y = -1/3
Substituting y = -1/3 into x - 3y = 2/3, we get x = 1.
Therefore, the solution to the system of equations is (x, y) = (1, -1/3).
CRAMER'S RULE
involves using determinants to solve for each variable in the system of equations.
For example, consider the system of equations: 2x + 3y = 7 4x - 2y = 10
We can write the system of equations as:
2x + 3y = 7
4x - 2y = 10
We can then represent the coefficients of the variables and the matrix of constants as matrices:
A = [ 2 3 ]
[ 4 -2 ]
B = [ 7 ]
[ 10]
Next, we can find the determinant of the coefficient matrix A:
|A| = 2*(-2) - 3*4 = -16
We can then find the determinant of the matrix formed by replacing the first column of A with the matrix of constants B:
|x| = [ 7 3 ]
[ 10 -2 ]
|x| = 7*(-2) - 3*10 = -44
Similarly, we can find the determinant of the matrix formed by replacing the second column of A with B:
|y| = [ 2 7 ]
[ 4 10]
|y| = 2*10 - 7*4 = -18
Finally, we can solve for x and y:
x = |x| / |A| = -44 / -16 = 11/4
y = |y| / |A| = -18 / -16 = 9/8
Therefore, the solution to the system of equations is (x, y) = (11/4, 9/8).
MATRIX METHOD
involves using matrix algebra to represent the coefficients of the variables in the system of equations. The system of equations can then be expressed as a matrix equation and solved using matrix operations.
For example, consider the system of equations: 2x + 3y = 7 4x - 2y = 10
[ 2 3 ] [ x ] [ 7 ]
[ 4 -2 ] [ y ] = [ 10]
First, we need to find the inverse of the coefficient matrix:
[ 2 3 ]^-1 = 1/16 [-2 -3]
[ -4 2]
Then, we can solve for x and y:
[ x ] [ 2 3 ]^-1 [ 7 ] [ 1 ]
[ y ] = [ 4 -2 ] [ 10] = [ -1/2 ]
Therefore, the solution to the system of equations is (x, y) = (1, -1/2).
MATRICES
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is a rectangular array of
numbers or other mathematical
objects, arranged in rows and
columns.
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are used to represent and manipulate systems of linear equations, and to perform other operations
such as linear transformations and determinants
TYPES OF MATRICES
SQUARE MATRIX
is a matrix with the same number of rows and columns. In other words, it is a matrix where the number of rows is equal to the number of columns.
a 3x3 matrix is a square matrix.
COLUMN MATRIX
is a matrix that has only one column. It is also known as a column vector or simply a vector.
[ 2 ]
[ 3 ]
[ 5 ]
DIAGONAL MATRIX
is a square matrix where all the non-diagonal elements are zero.
[ 2 0 0 ]
[ 0 5 0 ]
[ 0 0 -1 ]
ROW MATRIX
is a matrix that has only one row, while a column matrix is a matrix that has only one column.
[ 1 2 3 ]
MATHEMATICAL MODELING
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is the process of using mathematical concepts, tools, and techniques to represent real-world phenomena, systems, or processes.
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can be used in a wide range of fields, including physics, engineering, economics, biology, and social sciences. They are used to study complex systems that are difficult to understand or manipulate experimentally, or to make predictions about the behavior of a system in the future.
