Polynomial Operations 09.12

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Example: $$2x^3 - 4x^2 + 3x - 5$$ .

Simplifying a polynomial expression means to combine like terms and reduce the expression to its simplest form.

The square of a binomial is given by the formula: $$(a + b)^2 = a^2 + 2ab + b^2$$. Example: $$(x - 3)^2 = x^2 - 6x + 9$$.

To multiply a polynomial by a monomial, distribute the monomial to each term of the polynomial. Example: $$2x(-2x - 3) = -4x^2 - 6x$$.

The difference of two polynomials is found by subtracting the second polynomial from the first. Example: $$(3 - 2x + 2x^2) - (4x - 5 + 3x^2) = -x^2 - 6x + 8$$.

The sum of two polynomials is found by adding the corresponding coefficients of like terms. Example: $$(3x - 2x^2 + 3x^3 - 2) + (3x^2 - 2x - 9) = 3x^3 + x^2 + x - 11$$.

The product of a binomial squared is calculated using the formula: $$(a + b)^2 = a^2 + 2ab + b^2$$. Example: $$(2x + 3)^2 = 4x^2 + 12x + 9$$.

Like terms are terms that have the same variable raised to the same power. Example: $$3x^2$$ and $$5x^2$$ are like terms.

The degree of a polynomial is the highest power of the variable in the expression. Example: In $$2x^3 - 4x^2 + 3$$, the degree is 3.

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. Example: In $$4x^3 + 2x^2 - 5$$, the leading coefficient is 4.