Week 1 Advanced Algebra Concepts

Advanced algebra is a branch of mathematics that goes beyond basic algebraic operations to explore more complex structures and equations. It builds on fundamental concepts such as linear equations, quadratic equations, and polynomials, extending to topics like matrices, functions, and complex numbers.

Polynomial Functions

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. A general polynomial function of degree $n$ can be written as:
$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

Where $a_{n} \neq 0$ . The degree of the polynomial determines its behavior and the number of possible real roots.

Example:

$P (x) = 2 x^{3} - 5 x^{2} + 3 x - 7$

This is a cubic polynomial (degree 3), which can have up to 3 real roots.

Quick Check

Find all real roots of the polynomial $P(x) = x^3 - 4x^2 - 7x + 10$ .

1, 2, 5: Incorrect
-1, 2, 5: Correct
-2, -1, 5: Incorrect
-2, 1, 5: Incorrect

Solving Systems of Equations

In advanced algebra, solving systems of linear equations using methods like substitution, elimination, and matrix operations is essential. Consider the following system:

${\begin{cases} 2 x + 3 y = 5 \\ 4 x - y = 11 \end{cases}$

Matrix Representation:
This system can be written as $AX = B$ , where:

$A = [\begin{array}{cc} 2 & 3 \\ 4 & - 1 \end{array}], X = [\begin{array}{c} x \\ y \end{array}], B = [\begin{array}{c} 5 \\ 11 \end{array}]$

The solution can be found using the inverse of matrix $A$ , if it exists:
$X = A^{- 1} B$

Quick Check

Solve the system of equations:

$\begin{cases} 3x - y = 7 \\ 5x + 2y = 1 \end{cases}$

x = 1, y = -2: Correct
x = 2, y = 1: Incorrect
x = -1, y = 3: Incorrect
x = 3, y = -2: Incorrect

Complex Numbers

Complex numbers extend the real number system by introducing the imaginary unit $i$ , where $i^{2} = - 1$ . Every complex number is of the form $a + b i$ , where $a$ is the real part, and $bi$ is the imaginary part.

Operations with Complex Numbers:

Addition: $(a + b i) + (c + d i) = (a + c) + (b + d) i$
Multiplication: $(a + b i) (c + d i) = a c + a d i + b c i + b d i^{2}$
Since $i^{2} = - 1$

Quick Check

Multiply $(3 + 2i)$ and $(1 - 4i)$ . Simplify your answer.

$- 5 + 10 i$: Incorrect
$11 + 14 i$: Incorrect
$- 5 - 14 i$: Correct
$11 - 10 i$: Incorrect

Logarithmic and Exponential Equations

Logarithmic and exponential functions are inverses of each other and appear frequently in advanced algebra. Solving equations involving these functions often requires logarithmic identities:

$\log (a b) = \log a + \log b$
$\log (\frac{a}{b}) = \log a - \log b$
$\log (a^{n}) = n \log a$

Example: Solve $2^x = 16$
Solution: Rewrite 16 as $2^{4}$ , so $2^{x} = 2^{4}$ . Therefore, $x = 4$ .

Quick Check

Solve the equation $5^{x+1} = 125$ .

x = 1: Incorrect
x = 2: Correct
x = 3: Incorrect
x = 0: Incorrect

Conclusion

Advanced algebra is foundational for higher-level mathematics and applications in science, engineering, and computer science. Mastery of these concepts opens doors to understanding calculus, linear algebra, and differential equations. Practice and exploration are key to developing a deep understanding of these topics.