AP Precalculus: 1.4-1.6 Polynomials

Steps to identifying both the leading coefficient and the degree of a polynomial include…

1. Write the function in standard form (highest to lowest exponent).

2. Leading coefficient: Look at the coefficient of the term wth the highest exponent.

3. Degree: Look at the value of the highest exponent.

What is the leading coefficient and degree of this polynomial?: $x^2+5x^4-10$

leading coefficient of 5, degree of 4

What is the leading coefficient and degree of this polynomial?: $-7x^5 + 2x^3 - x + 12$

leading coefficient of $-7$, degree of $5$

What is the leading coefficient and degree of this polynomial?: $8 - 3x^2 + 4x^6$

leading coefficient of $4$, degree of $6$

Local/relative maximums occur...

Whenever the graph switches from increasing to decreasing

Local/relative minimums occur…

Whenever the graph switches from decreasing to increasing

If a function is changing from increasing to decreasing at x = d, then there is a(n) ____ at x = d.

local/relative maximum

If a function is changing from decreasing to increasing at x = d, then there is a(n) ____ at x = d.

local/relative minimum

Global/absolute maximums occur…

Whenever the function is at its highest point

Global/absolute minimums occur…

Whenever the function is at its lowest point

If the rate of change is positive on an interval, then the function is ____.

increasing

If the rate of change is negative on an interval, then the function is ____.

decreasing

If the rate of change is decreasing on an interval, then the function is ____.

concave down

If the rate of change is increasing on an interval, then the function is ____.

concave up

How do you find the average rate of change of a function?

If you are given an interval, plug in both values into the equation in order to find the y-values. Then, use the slope formula.

Find the average rate of change given $f(x) = 2^x + 4$ on the interval $[1, 3.5]$. Round your answer to 3 decimal places.

AROC: $3.725$

Find the average rate of change given $f(x) = 4^x - 1$ on the interval $[2, 3]$. Round your answer to 3 decimal places.

AROC: $48$

Find the average rate of change given $f(x) = 3^x + 2$ on the interval $[0.5, 2]$. Round your answer to 3 decimal places.

AROC: $4.845$

Factor: $x^2+2x-3$

$$\left(x+3\right)\left(x-1\right)$$

Factor: $x^3 + 3x^2 + 2x + 6$

$$\left(x^2 + 2\right)\left(x + 3\right)$$

In relation to polynomials, the fundamental theory of algebra states that…

A polynomial with degree n, has exactly n roots.

How many roots (real and complex) does this polynomial function have?: $100x^7-847x^6+29x^3-2\pi x+6$

$$7$$

What are complex zeros?

Complex zeros, a.k.a complex roots, are any zeros of a function that contains the imaginary number, i. Complex zeros come in conjugate pairs. Conjugate pairs have opposite imaginary number signs. (Ex: 4i and -4i)