A polynomial is an algebraic expression consisting of variables raised to non-negative integer powers and multiplied by constant coefficients. Polynomials are generally written in the form:

𝑃(𝑥) = 𝑎_𝑛𝑥^𝑛 + 𝑎_{𝑛−1}𝑥^{𝑛−1} + ⋯ + 𝑎_1𝑥 + 𝑎_0

Where:
• 𝑎_𝑛,𝑎_{𝑛−1},…,𝑎_1,𝑎_0 are constants (coefficients),
• 𝑛 is a non-negative integer (degree of the polynomial),
• 𝑥 is the variable.

Types of Polynomials:
• Monomial: A polynomial with only one term (e.g., 4𝑥^3).
• Binomial: A polynomial with two terms (e.g., 𝑥^2 + 2𝑥).
• Trinomial: A polynomial with three terms (e.g., 𝑥^2 + 5𝑥 + 6).

Degree of a Polynomial: The degree of a polynomial is the highest power of the variable in the expression. For example, the degree of 4𝑥^3 + 3𝑥^2 − 𝑥 + 7 is 3.

Zero of a Polynomial: The zero or root of a polynomial is the value of 𝑥 for which the polynomial equals zero. For example, if 𝑃(𝑥) = 𝑥^2 − 4, the zeros of the polynomial are 𝑥 = 2 and 𝑥 = −2.

The Remainder Theorem states that if a polynomial 𝑃(𝑥) is divided by a linear divisor 𝑥−𝑐, then the remainder of the division is 𝑃(𝑐).

Mathematically:
𝑃(𝑥) = (𝑥 − 𝑐)𝑄(𝑥) + 𝑅
Where 𝑄(𝑥) is the quotient, and 𝑅 is the remainder. According to the Remainder Theorem, 𝑅 = 𝑃(𝑐).

Example:
For 𝑃(𝑥) = 𝑥^3 − 3𝑥^2 + 2𝑥 − 5, if we divide by 𝑥 − 2, then the remainder is 𝑃(2).
𝑃(2) = (2)^3 − 3(2)^2 + 2(2) − 5 = 8 − 12 + 4 − 5 = −5
Thus, the remainder is −5.

The Factorization Theorem states that if 𝑥−𝑐 is a factor of the polynomial 𝑃(𝑥), then 𝑃(𝑐) = 0. In other words, the remainder of dividing 𝑃(𝑥) by 𝑥−𝑐 is zero.

Example:
If 𝑃(𝑥) = 𝑥^3 − 3𝑥^2 + 2𝑥 − 6, and 𝑥 − 2 is a factor, then by the Factorization Theorem, 𝑃(2) = 0. To find the complete factorization of 𝑃(𝑥), divide 𝑃(𝑥) by 𝑥 − 2.

Algebraic identities are equations that hold true for all values of the variables involved. Some of the key identities to remember are:
• Square of a Binomial: (𝑎 + 𝑏)² = 𝑎² + 2𝑎𝑏 + 𝑏²
• Difference of Squares: 𝑎² − 𝑏² = (𝑎 + 𝑏)(𝑎 − 𝑏)
• Cube of a Binomial: (𝑎 + 𝑏)³ = 𝑎³ + 3𝑎²𝑏 + 3𝑎𝑏² + 𝑏³
• Sum or Difference of Cubes: 𝑎³ + 𝑏³ = (𝑎 + 𝑏)(𝑎² − 𝑎𝑏 + 𝑏²), 𝑎³ − 𝑏³ = (𝑎 − 𝑏)(𝑎² + 𝑎𝑏 + 𝑏²)

These identities are used frequently in simplifying algebraic expressions, solving equations, and factoring polynomials.

A quadratic equation is an equation of the form:
𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0
Where 𝑎, 𝑏, and 𝑐 are constants, and 𝑎 ≠ 0.

The solutions to quadratic equations can be found using the Quadratic Formula:
−𝑏 ± √(𝑏² − 4𝑎𝑐)
𝑥 = _____
2𝑎
This formula gives two solutions (roots), which are the values of 𝑥 that satisfy the quadratic equation.

Example:
For 2𝑥² − 4𝑥 − 6 = 0, using the quadratic formula:
−(−4) ± √((−4)² − 4(2)(−6))
4 ± √(16 + 48)
4 ± √64
4 ± 8
𝑥 = _ = 3 and 𝑥 = _ = −1
4
4

Algebraic equations may also involve fractions and radicals. These types of equations can be solved by various methods, such as:
• Multiplying both sides by the denominator (to eliminate fractions),
• Squaring both sides (to eliminate radicals),
• Using substitution or other techniques.

Example:
For the equation 1/x + 3 = 5, multiply through by x to get rid of the fraction:
1 + 3𝑥 = 5𝑥
Solving for 𝑥:
1 = 2𝑥 ⇒ 𝑥 = ½.

Simultaneous equations involve solving two or more equations that are linked by common variables. These equations can be solved by methods such as:
• Substitution Method
• Elimination Method
• Graphical Method

Example:
Solve the system of equations:
𝑥 + 𝑦 = 7
𝑥 − 𝑦 = 3
By adding the two equations:
(𝑥 + 𝑦) + (𝑥 − 𝑦) = 7 + 3
2𝑥 = 10 ⇒ 𝑥 = 5
Substitute 𝑥 = 5 into 𝑥 + 𝑦 = 7:
5 + 𝑦 = 7 ⇒ 𝑦 = 2.
Thus, the solution is 𝑥 = 5 and 𝑦 = 2.