Naming Polynomials and Polynomial Operations

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, $$2x^2 + 3x - 5$$ is a polynomial.

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $$4x^3 + 2x^2 - x$$ , the degree is 3.

Polynomials can be classified as: - Constant (degree 0) - Linear (degree 1) - Quadratic (degree 2) - Cubic (degree 3) - Quartic (degree 4), etc.

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

To add polynomials, combine like terms. For example, $$(2x^2 + 3x) + (4x^2 - 2x) = (2x^2 + 4x^2) + (3x - 2x) = 6x^2 + x$$ .

To subtract polynomials, distribute the negative sign and then combine like terms. For example, $$(5x^2 + 3) - (2x^2 + 1) = (5x^2 - 2x^2) + (3 - 1) = 3x^2 + 2$$ .

To multiply polynomials, use the distributive property (FOIL for binomials) and combine like terms. For example, $$(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$$ .

To multiply a polynomial by a monomial, distribute the monomial to each term in the polynomial. For example, $$4a(5a^2 - a + 4) = 20a^3 - 4a^2 + 16a$$ .

The perimeter P is given by the formula $$P = 2(length + width)$$ . Thus, $$P = 2((2x - 4) + (2x + 3)) = 2(4x - 1) = 8x - 2$$ .

The expression $$m^2 - 25$$ can be factored as $$(m - 5)(m + 5)$$ , which is a difference of squares.