Factor ƒ(x)ƒ(x) into linear factors given that k is a zero. ƒ(x)=2x4+x3−9x2−13x−5; k=−1ƒ(x)=$$2x^4+x^3$-9x^2-13x-5$$;\text{ }k=-1 (multiplicity 33)

Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 32

4장, 문제 32

Factor $ƒ (x)$ into linear factors given that k is a zero. $ƒ (x) = 2 x^{4} + x^{3} - 9 x^{2} - 13 x - 5; k = - 1$ (multiplicity $3$ )

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Since $ k = -1 $ is a zero of multiplicity 3, it means $ (x + 1)^3 $ is a factor of $ f(x) $. Start by dividing $ f(x) = 2x^4 + x^3 - 9x^2 - 13x - 5 $ by $ (x + 1)^3 $ to find the remaining factor.

First, perform polynomial division or synthetic division to divide $ f(x) $ by $ (x + 1) $ three times, each time reducing the degree of the polynomial and confirming the zero at $ x = -1 $.

After dividing by $ (x + 1)^3 $, you will obtain a quotient polynomial of degree 1 (linear factor). Write $ f(x) $ as $ (x + 1)^3 $ times this quotient.

Express the quotient polynomial in standard linear form $ ax + b $. This will complete the factorization of $ f(x) $ into linear factors.

Finally, write the complete factorization as $ f(x) = 2 (x + 1)^3 (ax + b) $, where $ a $ and $ b $ come from the quotient found in the division steps.

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주요 개념

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

Polynomial Zeros and Multiplicity

A zero of a polynomial is a value of x that makes the polynomial equal to zero. Multiplicity refers to how many times a particular zero is repeated as a root. For example, if k = -1 is a zero with multiplicity 3, then (x + 1) is a factor repeated three times in the factorization.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a factor corresponding to a known zero. Synthetic division is a shortcut method for dividing by linear factors like (x - k). Repeated division by (x + 1) three times will reduce the polynomial step-by-step, helping to factor it completely.

Factoring Polynomials into Linear Factors

Factoring a polynomial into linear factors means expressing it as a product of first-degree polynomials. After dividing out the repeated factor (x + 1) three times, the remaining polynomial can be factored further or solved for zeros to complete the factorization into linear terms.

Factor ƒ(x)ƒ(x) into linear factors given that k is a zero. ƒ(x)=2x4+x3−9x2−13x−5; k=−1ƒ(x)=2x^4+x^3-9x^2-13x-5;\(\text{ }\)k=-1 (multiplicity 33)