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 14

4장, 문제 14

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x³+x²−3x+1

검증된 단계별 안내

Identify the polynomial function:

f (x)=2x^3+x^2-3x+1

List all possible rational zeros using the Rational Root Theorem. The possible rational zeros are of the form

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

, where

p

divides the constant term (1) and

q

divides the leading coefficient (2). So, possible rational zeros are

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

Use synthetic division to test each possible rational zero. Start by testing

x=1

: set up synthetic division with coefficients

2, 1, -3, 1

and divide by

x-1

. Check the remainder to see if it is zero, indicating a root.

Once you find a zero (say

r

), use the quotient polynomial from the synthetic division (which will be a quadratic) to find the remaining zeros.

Solve the quadratic quotient either by factoring, completing the square, or using the quadratic formula to find the remaining zeros of the polynomial.

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.

Rational Root Theorem

The Rational Root Theorem helps identify all possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. These possible roots are expressed as ±(factors of constant term)/(factors of leading coefficient), providing a finite list to test.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form (x - c). It simplifies calculations to test whether a candidate root is an actual zero by checking if the remainder is zero, and it produces a quotient polynomial for further analysis.

Factoring Polynomials and Finding Zeros

Once a zero is found using synthetic division, the quotient polynomial can be factored further or solved using other methods to find remaining zeros. This step breaks down the polynomial into simpler factors, revealing all roots including real and complex zeros.

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x3+x2−3x+1