Polynomials and Volume

• Polynomials are algebraic expressions containing variables and coefficients, that only use addition, subtraction, multiplication, and non-negative integer exponents.
• Polynomials are either a monomial (one term) or the sum of monomials.
• (3x^2 - 2x + 1), (5y^3 + 2y - 7), and (8z^4) are Polynomial examples.
• These are not polynomials: (3x^{-2}) (negative exponent), (2\sqrt{x}) (fractional exponent), and (1/(x+1)) (variable in the denominator).
• Polynomial degree is the variable's highest power.
• Leading coefficient is the coefficient of the highest degree term.

Polynomial Operations

• Addition and Subtraction: Like terms are combined, with the same variable and exponent.
• Multiplication uses the distributive property to multiply each term in one polynomial by each term in the other polynomial.
• Division can be performed using polynomial long division or synthetic division.

Volume

• Volume measures 3D space.
• Volume is measured in cubic units, such as (cm^3), (m^3), (in^3), or (ft^3).

Volume Formulas - Prisms

• A prism is a solid geometric figure with two similar, equal, and parallel end faces, and sides that are parallelograms.
• Prism Volume = (Base Area) × (height), shown as (V = Bh), where (B) is the base area and (h) is prism height.
• Rectangular Prism Volume: (V = lwh), where (l) is length, (w) is width, and (h) is height.
• Cube Volume: (V = s^3), where (s) is the side length.
• Triangular Prism Volume: (V = (1/2 \cdot b \cdot h') \cdot h), where (b) is the triangle's base, (h') is triangle height, and (h) is prism height.

Volume Formulas - Cylinders and Cones

• A cylinder is a solid geometric figure with straight parallel sides and a circular or oval section.
• Cylinder Volume: (V = \pi r^2 h), where (r) is base radius and (h) is height.
• A cone is a solid geometric figure with a circular base and a pointed tip.
• Cone Volume: (V = (1/3) \pi r^2 h), where (r) is base radius and (h) is height.

Volume Formulas - Spheres

• A sphere is a perfectly round geometrical object in three-dimensional space.
• Sphere Volume: (V = (4/3) \pi r^3), where (r) is the radius.

Surface Area

• Surface area is the total area of all the surfaces of a three-dimensional object.
• Surface area is measured in square units like (cm^2), (m^2), (in^2), and (ft^2).

Surface Area Formulas - Prisms

• Rectangular Prism Surface Area: (SA = 2lw + 2lh + 2wh), where (l) is length, (w) is width, and (h) is height.
• Cube Surface Area: (SA = 6s^2), where (s) is the side length.
• Triangular Prism Surface Area: (SA = bh + 2ls + lb), where (b) is the triangle's base, (h) is the triangle's height, (l) is the prism's length, and (s) is the triangle's side length.

Surface Area Formulas - Cylinders and Cones

• Cylinder Surface Area: (SA = 2\pi r^2 + 2\pi rh), where (r) is the radius and (h) is the height.
• Cone Surface Area: (SA = \pi r^2 + \pi r s), where (r) is the radius and (s) is the slant height ((s = \sqrt{r^2 + h^2})).

Surface Area Formulas - Spheres

• Sphere Surface Area: (SA = 4\pi r^2), where (r) is the radius.

Quadrilaterals

• A quadrilateral is a polygon that has four sides, four vertices, and four angles.
• The sum of the interior angles in any quadrilateral is 360 degrees.

Types of Quadrilaterals - Parallelogram

• A parallelogram has two pairs of parallel sides.
• Opposite sides are equal in length.
• Opposite angles are equal.
• Consecutive angles are supplementary (add up to 180 degrees).
• Diagonals bisect each other.
• Area: (A = bh), where (b) is the base and (h) is the height.

Types of Quadrilaterals - Rectangle

• A rectangle is a parallelogram that has four right angles.
• Its opposite sides are equal and parallel.
• Diagonals are equal in length and bisect each other.
• Area: (A = lw), where (l) is length and (w) is width.

Types of Quadrilaterals - Square

• A square is a rectangle that has four sides equal in length.
• All sides are equal and parallel.
• All angles are right angles.
• Diagonals are equal in length, bisect each other at right angles, and bisect its angles.
• Area: (A = s^2), where (s) is the side length.

Types of Quadrilaterals - Rhombus

• A rhombus is a parallelogram that has four sides equal in length.
• Its opposite sides are parallel.
• Opposite angles are equal.
• Diagonals bisect each other at right angles.
• Diagonals bisect the angles of the rhombus.
• Area: (A = (1/2)d_1d_2), where (d_1) and (d_2) are the lengths of the diagonals.

Types of Quadrilaterals - Trapezoid

• A trapezoid has at least one pair of parallel sides.
• The parallel sides are called bases, and the non-parallel sides are called legs.
• An Isosceles Trapezoid has legs that are of equal length.
• Base angles are equal in an isosceles trapezoid.
• Area: (A = (1/2)(b_1 + b_2)h), where (b_1) and (b_2) are the lengths of the bases and (h) is the height.

Types of Quadrilaterals - Kite

• A kite has two pairs of adjacent sides that are equal in length.
• Diagonals are perpendicular.
• One diagonal bisects the other diagonal.
• One diagonal bisects a pair of opposite angles.
• Area: (A = (1/2)d_1d_2), where (d_1) and (d_2) are the lengths of the diagonals.