Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 57

4장, 문제 57

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=2x⁴−3x³−7x²−8x+6

검증된 단계별 안내

Start by finding the zeros of the polynomial function

f (x) = 2 x^{4} - 3 x^{3} - 7 x^{2} - 8 x + 6

by setting

f (x) = 0

and solving for

x

Use the Rational Root Theorem to list possible rational zeros. These are of the form

\(\pm\) \(\frac{p}{q}\)

, where

p

divides the constant term 6 and

q

divides the leading coefficient 2. So possible roots are

\(\pm\) 1, \(\pm\) 2, \(\pm\) 3, \(\pm\) 6, \(\pm\) \(\frac{1}{2}\), \(\pm\) \(\frac{3}{2}\)

Test each possible rational root by substituting into the polynomial or using synthetic division to check if it yields zero. When a root is found, factor it out from the polynomial.

After factoring out the first root, reduce the polynomial degree and repeat the process to find other zeros. Continue factoring until the polynomial is completely factored into linear and/or irreducible quadratic factors.

Once all zeros are found, analyze the multiplicity of each zero to understand the behavior of the graph at those points. Use this information along with the leading coefficient and degree to sketch the complete graph without a graphing utility.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Finding Zeros of Polynomial Functions

Zeros of a polynomial are the values of x for which the function equals zero. To find them, one can use factoring, synthetic division, or the Rational Root Theorem to test possible roots. Identifying all zeros is essential for understanding the behavior and shape of the graph.

End Behavior of Polynomial Functions

The end behavior describes how the function behaves as x approaches positive or negative infinity. It depends on the leading term's degree and coefficient. For example, a positive leading coefficient with an even degree means the graph rises on both ends, guiding the sketching of the complete graph.

Sketching Polynomial Graphs Without Technology

Drawing a polynomial graph by hand involves plotting zeros, determining multiplicities, analyzing end behavior, and finding key points like local maxima and minima. Understanding these features helps create an accurate, complete graph without relying on graphing utilities.

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=2x4−3x3−7x2−8x+6