Podcast
Questions and Answers
What is the polynomial function in standard form with the written zeros -5, -3, and 1?
What is the polynomial function in standard form with the written zeros -5, -3, and 1?
What are the zeros of the function y = x³ - 9x?
What are the zeros of the function y = x³ - 9x?
0, -3, 3
What is the cubic function after being shifted right 2, down 10, and reflected across the x-axis?
What is the cubic function after being shifted right 2, down 10, and reflected across the x-axis?
y = -(x-2)³
Factor the expression 3x³ - 21x² + 30x completely.
Factor the expression 3x³ - 21x² + 30x completely.
Signup and view all the answers
Factor the expression x³ + 64 completely.
Factor the expression x³ + 64 completely.
Signup and view all the answers
Factor the expression x⁴ - 41x² + 400 completely.
Factor the expression x⁴ - 41x² + 400 completely.
Signup and view all the answers
Factor the expression x³ - 7x² - 324x + 2268 completely.
Factor the expression x³ - 7x² - 324x + 2268 completely.
Signup and view all the answers
Find P(4) for P(x) = x⁴ - 3x³ + 3x² + 6x + 3.
Find P(4) for P(x) = x⁴ - 3x³ + 3x² + 6x + 3.
Signup and view all the answers
Given the volume of a suitcase represented by the polynomial x³ + 7x² + 4x - 12, with height (x-1), find the length assuming it is greater than the width.
Given the volume of a suitcase represented by the polynomial x³ + 7x² + 4x - 12, with height (x-1), find the length assuming it is greater than the width.
Signup and view all the answers
Study Notes
Polynomial Functions and Zeros
- A polynomial function can be expressed in standard form using its zeros. For zeros -5, -3, and 1, the function becomes ( (x+5)(x+3)(x-1) ).
- To expand, use the product of the first two brackets and then multiply by the last term, resulting in ( x^3 + 7x^2 + 7x - 15 ).
Finding Zeros of a Function
- The function ( y = x^3 - 9x ) is factored into ( x(x^2 - 9) ), which further simplifies to ( x(x+3)(x-3) ).
- The zeros of the function are determined by setting each factor to zero: ( x = 0 ), ( x = -3 ), and ( x = 3 ).
Function Transformations
- A cubic function can be transformed through shifts and reflections. For example, shifting right by 2, down by 10, and reflecting across the x-axis results in the equation ( y = -(x-2)^3 ).
Factoring Polynomials Completely
- For the polynomial ( 3x^3 - 21x^2 + 30x ), factor out ( 3x ), yielding ( 3x(x^2 - 7x + 10) ). This further factors to ( 3x(x-5)(x-2) ).
- The expression ( x^3 + 64 ) is a sum of cubes and can be factored using the formula ( (a^3 + b^3) = (a+b)(a^2 - ab + b^2) ), leading to ( (x+4)(x^2 - 4x - 16) ).
Advanced Factoring Techniques
- The polynomial ( x^4 - 41x^2 + 400 ) can be factored by recognizing it as a quadratic in terms of ( x^2 ), leading to ( (x^2 - 25)(x^2 - 16) ) and final factors of ( (x-5)(x+5)(x-4)(x+4) ).
- A more complex expression like ( x^3 - 7x^2 - 324x + 2268 ) can be split into two groups for factoring, ultimately resulting in ( (x-18)(x+18)(x-7) ).
Evaluating Polynomials
- To evaluate ( P(4) ) for ( P(x) = x^4 - 3x^3 + 3x^2 + 6x + 3 ), substitute ( x ) with 4 and compute: ( 4^4 - 3(4^3) + 3(4^2) + 6(4) + 3 ), resulting in ( R = 475 ).
Volume and Polynomial Division
- For a rectangular suitcase, the polynomial representing volume is ( x^3 + 7x^2 + 4x - 12 ) with height ( (x-1) ). Synthetic division is used to divide and factor it, yielding ( (x-1)(x+6)(x+2) ).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of polynomial functions with these flashcards. Cover essential concepts like writing polynomials in standard form and finding zeros of functions. Perfect for Algebra 2 students preparing for Unit 6.
