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Questions and Answers

What is a polynomial function?

A function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation.

What is a polynomial expression?

X² - 5x + 6

The highest exponent in a polynomial is called the degree.

True

What are the roots of the polynomial equation X² - 5x + 6 = 0?

2 and 3 Signup and view all the answers

What is the standard form of a polynomial function?

The polynomial function is in standard form when the terms are arranged from the highest exponent down to the lowest. Signup and view all the answers

What is the degree of the polynomial (X - 2) (X + 1) (X - 5) = 0?

3 Signup and view all the answers

What are the roots of the polynomial equation (X - 2) (X + 1) (X - 5) = 0?

2, -1, 5 Signup and view all the answers

What is the degree of the polynomial equation (x + 1)³ (x - 2)² (x + 1) = 0?

6 Signup and view all the answers

Which of the following is true about the roots of a polynomial equation written in factor form?

The roots are the opposite sign of the numbers in the factors. Signup and view all the answers

How do you find the polynomial equation given the roots?

Multiply the factors together, where each factor is written as (x - root). Signup and view all the answers

What does a polynomial equation's x-intercepts represent?

The roots of the polynomial equation Signup and view all the answers

The graph of an odd-degree polynomial function always rises on both sides.

False Signup and view all the answers

The graph of an even-degree polynomial function always falls on both sides.

False Signup and view all the answers

Which of the following describes the graph of an odd-degree polynomial function with a positive leading coefficient?

Falls to the left, rises to the right Signup and view all the answers

Which of the following describes the graph of an even-degree polynomial function with a negative leading coefficient?

Falls to the left, falls to the right Signup and view all the answers

What is the difference between a chord and a diameter in a circle?

A chord is any line segment connecting two points on the circle's circumference, while a diameter is a specific chord that passes through the center of the circle. Signup and view all the answers

What is the relationship between the radius, diameter, and circumference of a circle?

The diameter is twice the length of the radius. The circumference is the length of the circle's boundary, and it's calculated using the formula C = πd, where C is the circumference, d is the diameter, and π is a mathematical constant approximately equal to 3.14. Signup and view all the answers

Match the following terms with their descriptions:

Radius = A line segment connecting two points on the circumference Diameter = The distance around the circle Chord = The segment from the center of the circle to a point on the circumference Circumference = The line segment connecting two points on the circumference and passing through the center Arc = A section of the circle's circumference Sector = The region enclosed by two radii and the arc between them Segment = The region enclosed by a chord and the arc between its endpoints Signup and view all the answers

What is the distance formula?

d = √(x2 - x1)² + (y2 - y1)² Signup and view all the answers

The midpoint of a line segment is always the same distance away from both endpoints.

True Signup and view all the answers

The distance formula can be used to find the length of any line segment, not just horizontal or vertical ones.

True Signup and view all the answers

Study Notes

Polynomials

Polynomials are functions involving only non-negative integer powers or positive integer exponents of variables.
Examples of polynomial functions include f(x) = x² - 5x + 6, and y = x² - 5x + 6
Polynomial expressions are algebraic expressions with multiple terms containing variables and their powers, like x² - 5x + 6
Polynomial equations set a polynomial expression equal to zero, such as x² - 5x + 6 = 0.
When a polynomial function is in standard form, the exponents are ordered from highest to lowest.
The highest exponent determines the degree of the polynomial.

Factor Form

Factor form represents a polynomial as a product of factors.
Examples of factor form include (x - 2) (x + 1) (x - 5) = 0 and (x + 1)³ (x - 2)² (x + 1) = 0
The degree of a polynomial in factor form is the sum of the exponents in the factors.
Roots/x-intercepts are the values of x where the polynomial equals zero, and their signs are the opposite in the factor form.

Finding Polynomial Equations given Roots

To find a polynomial equation given roots, use factors in the form (x - root).
Example: If the roots are -1, 3 , 2 , the factor form of the equation is (x + 1)(x - 3)(x - 2) = 0.
Expand to get the full polynomial equation.
Example, for the roots -1, 3 and 2 the full polynomial equation becomes: x³ - 4x² + x + 6 = 0

X and Y intercepts

X-intercepts: The points where the graph of a polynomial function crosses the x-axis. They are found when y = 0.
Y-intercepts: The point where the graph of a polynomial function crosses the y-axis. They are found when x = 0.

Graphing Polynomial Functions

The degree and the sign of the leading coefficient of a polynomial determine its end behavior (whether it rises or falls to the left and right)
Odd degree with positive leading coefficient -> Falls left, rises right
Odd degree with negative leading coefficient -> Rises left, falls right
Even degree with positive leading coefficient -> Rises left, rises right
Even degree with negative leading coefficient -> Falls left, falls right.

Circle

A circle is a set of points equidistant from a central point (the center).
Parts of a circle:
- Center: Fixed point, equidistant from all points
- Radius: Distance from center to a point on the circle.
- Diameter: Distance across the circle through the center.
- Circumference: Distance around the whole circle.
- Arc: Part of the circumference
- Chord: A line segment connecting two points on a circle.
- Secant: A line that intersects the circle at two points.
- Tangent: A line that touches the circle at exactly one point.
- Sector: Region bounded by two radii and an arc.
- Segment: Region bounded by a chord and an arc.

Theorems on Secant Segments, Tangent Segments, and External Secant Segments

Theorems relate products of lengths from exterior points to the lengths of various segments on circles.
Secant segments and tangent segments from exterior points involve relationships that relate external parts of secants.