Algebraic Expressions, Equations, and Identities

Posted on Feb 27, 2025 in Mathematics

Algebraic Expressions

An algebraic expression is a combination of numbers and letters connected by arithmetic operation signs. The letters are called variables.

Monomials

A monomial is an algebraic expression where the only operations affecting the variables are multiplication and natural exponentiation. Monomials with the same literal part are like terms.

Polynomials

A polynomial is an algebraic expression formed by the sum or difference of two or more monomials. The degree of a polynomial is the highest degree of the monomials that form it.

Division of Polynomials

A monomial is divisible by another if dividing one by the other results in a polynomial. To divide a polynomial by a monomial, divide each of its terms by the monomial divisor. A polynomial is divisible by a monomial if the quotient obtained is another polynomial.

Ruffini’s Rule

Ruffini’s rule is a short form to divide a polynomial P(x) by a binomial of the form x – a (or x + a).

Remainder Theorem

The remainder when dividing a polynomial P(x) by x – a coincides with the numerical value of the polynomial for x = a, P(a). When dividing a polynomial P(x) by x – a, the remainder is 0 (equivalent to P(a) = 0). The division is exact, and we say that ‘a’ is a root of the polynomial. The possible roots of a polynomial are among the divisors of its independent term. The roots of a polynomial are the values of “x” for which the polynomial equals 0.

Identities and Equations

Identities

Identities are algebraic expressions where equality holds for any value of the variables.

Equations

Equations are algebraic expressions that are equalities between terms for which certain variables have known solutions. If two equations have the same solutions, we say they are equivalent.

Types of Equations

Equations of the 1st Degree

These are equations that can be reduced to the form ax = b (the exponent of the unknown is 1).

Equations of the 2nd Degree

These are equations that can be reduced to the form ax2 + bx + c = 0. When a, b, and c are distinct from zero, we say it is complete, and the equation is solved using the formula. The expression b2 – 4ac is called the discriminant of the equation of the 2nd degree.

  • When the discriminant is positive, the equation has two solutions.
  • When the discriminant is 0, the equation has 1 solution (double).
  • When the discriminant is negative, the equation has no solution.

When the coefficients are 0, we say the equation is incomplete and is resolved by extracting the common factor x.

Biquadratic Equations

These are equations that can be reduced to the form ax4 + bx2 + c = 0. To solve, a variable change is made: z = x2, z2 = x4; az2 + bz + c = 0. This equation of the 2nd degree is solved, and “z” is found. This also allows solving equations of the form: ax6 + bx3 + c = 0.

Radical Equations

These are equations where the unknown appears under a radical. To solve, isolate the radical on one of the two sides, then raise both sides of the equation to the power of the index of the radical and solve the resulting equation. After finding the solution, verify it.

Solving Systems of Equations

Substitution Method

  1. Uncover the unknowns of the system.
  2. In the other equation, substitute the unknown that was cleared.
  3. Solve the equation.
  4. Retrieve the value of the unknown, replacing its value in the first equation.

Reduction Method

  1. This system aims to eliminate one unknown by multiplying the equations by a determined number.
  2. Eliminate the unknown “x” to find the unknown “y” and substitute its value in any of the system equations.