Questions and Answers
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?
0, -3, 3
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.
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Factor the expression x³ + 64 completely.
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Factor the expression x⁴ - 41x² + 400 completely.
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Factor the expression x³ - 7x² - 324x + 2268 completely.
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Find P(4) for P(x) = x⁴ - 3x³ + 3x² + 6x + 3.
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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.
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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) ).
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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.
