Basics of Literal Algebra: Monomials and Polynomials

Flashcards covering definitions, properties, arithmetic operations, special products, polynomial long division, and Ruffini's method from Lesson 1: Basics of Literal Algebra.

How is a monomial defined in literal calculation?

A monomial is a set of letters and numbers linked together exclusively by multiplication and/or division operations.

What three parts are always recognizable in every monomial?

The three parts are: the sign ($+$ or $-$), the coefficient (the numeric part, typically including the sign), and the literal part (all letters appearing in the monomial written with their exponents).

In a monomial such as $a^3$, what is the numerical coefficient if none is written?

If the coefficient is not explicitly written, it is equal to $1$.

When are two or more monomials defined as being similar to each other?

Two or more monomials are similar when they have the exact same literal part.

How is the sum of two or more similar monomials calculated?

The sum is a monomial whose coefficient is the algebraic sum of the coefficients (each taken with its own sign) and whose literal part is identical to the literal part of the addend monomials.

What must be done when adding monomials that are not similar?

They cannot be combined; they must be rewritten without performing operations, which means the sum is "left indicated."

What effect does placing a negative sign before a subtrahend monomial have?

A negative sign placed before the subtrahend makes the sign of the subtrahend itself opposite.

Unlike addition and subtraction, what condition is required to multiply or divide monomials?

No condition is required; all monomials are indiscriminately multiplicable or divisible with each other.

What are the three steps followed to calculate the product of two or more monomials?

1) Multiply the signs of the monomials involved according to the sign rules. 2) Multiply the coefficients. 3) Multiply the literal parts by adding the exponents of identical letters.

What are the three steps followed to divide monomials?

1) Divide the signs according to the sign rules. 2) Divide the coefficients. 3) Divide the literal parts by subtracting the exponents of identical letters.

What are the rules for signs when raising a monomial to an exponent $n$?

Both a positive ($+$) and a negative ($-$) sign raised to an even $n$ equal positive ($+$); a positive ($+$) sign raised to an odd $n$ equals positive ($+$); and a negative ($-$) sign raised to an odd $n$ equals negative ($-$).

Who is the historical figure shown in the image, regarded as the "father" of algebra?

The astronomer al-Khuwarizmi, who lived between the 700s and 800s AD and founded the "House of Knowledge" in Baghdad.

From which historical mathematician's name is the word "algorithm" derived?

It is derived from the name of the astronomer al-Khuwarizmi.

What is a polynomial, and how are polynomials with two, three, or four terms named?

A polynomial is a set of monomials (termed "terms") related by addition and subtraction operators. A polynomial of two terms is called a binomial, three terms is a trinomial, and four terms is a quadrinomial.

What is the effect of a negative sign preceding a parenthesis containing a polynomial?

The negative sign has the effect of making opposite the signs of all the terms of the polynomial contained within the parenthesis.

How is the product of two polynomials computed?

It is computed by finding the sum of the products of each term of the first polynomial multiplied by each term of the second polynomial, following the distributive property.

What is the formula and rule for calculating a squared binomial $(A + B)^2$?

$(A + B)^2 = A^2 + 2AB + B^2$, which equals the first term squared plus the second term squared plus (or minus) twice the product of the first term and the second term.

What is the formula and rule for expanding a cubed binomial $(A + B)^3$?

$(A + B)^3 = A^3 + B^3 + 3A^2B + 3AB^2$, which equals the first term cubed plus (or minus) the second term cubed plus (or minus) the triple product of the first term squared by the second term plus (or minus) the triple product of the second term squared by the first term.

What is the rule and formula for expanding a squared polynomial such as $(A + B + C + D)^2$?

It is the sum of the squares of each term plus (or minus) the double products of each term and all the terms that follow it: $(A + B + C + D)^2 = A^2 + B^2 + C^2 + D^2 + 2AB + 2AC + 2AD + 2BC + 2BD + 2CD$.

According to Newton's Rule for developing $(A + B)^n$, how are the number of terms and subsequent coefficients determined?

It consists of $n + 1$ terms. The first coefficient is always $1$, the second is always $n$, and each subsequent coefficient is obtained by multiplying the coefficient of the previous term by the exponent of $A$ in that term and dividing by the exponent of $B$ increased by $1$.

What is the formula and rule for the product of the sum of two monomials by their difference?

$(A + B) \cdot (A - B) = A^2 - B^2$, which equals the first term squared minus the second term squared.

When dividing two polynomials using long division, how should incomplete polynomials be arranged in the grid?

They should be written in the grid taking into account the missing degree of the variable (leaving appropriate spaces for any missing powers).

What two conditions must the divisor satisfy in order to apply Ruffini's method directly?

The unknown term of the divisor polynomial must: 1) be first degree, and 2) have a coefficient of $1$.

In Ruffini's method, what is the degree of the quotient polynomial compared to the dividend?

The quotient polynomial starts from a lower degree (one degree lower) of the unknown letter than the dividend polynomial.

How can Ruffini's method be applied if the coefficient of the unknown letter in the divisor is not equal to $1$?

Divide both the terms of the dividend and the terms of the divisor by the coefficient of the divisor's unknown term, using the invariant property of fractions.

When dividing the dividend and divisor by a constant to apply Ruffini's method, what must be done to the resulting remainder?

The remainder obtained from the Ruffini grid must be remultiplied by the number by which the polynomials were previously divided.