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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 32
Chapter 4, Problem 32

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)

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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Finding Zeros & Their Multiplicity

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.
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Introduction to Factoring Polynomials

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.
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Related Practice
Textbook Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x3-37x2+50x+60 between 7 and 8

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=-x(x+1)(x-1)

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Textbook Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x3-37x2+50x+60 between 2 and 3

279
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Textbook Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x^3-37x^2+50x+60 Find the zero in part (b) to three decimal places.

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Textbook Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 Find the zero in part (b) to three decimal places.

365
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Textbook Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 between 2 and 3

369
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