Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 41
Capitolo 4, Problema 41

Use synthetic division to divide f(x)=x3−4x2+x+6 by x+1. Use the result to find all zeros of f.

Guida verificata passo dopo passo
1
Identify the divisor and rewrite it in the form \( x - c \). Since the divisor is \( x + 1 \), rewrite it as \( x - (-1) \), so \( c = -1 \).
Set up synthetic division by writing the coefficients of \( f(x) = x^3 - 4x^2 + x + 6 \) in order: \( 1, -4, 1, 6 \).
Perform synthetic division using \( c = -1 \): bring down the first coefficient, multiply by \( c \), add to the next coefficient, and repeat this process for all coefficients.
Write the quotient polynomial from the synthetic division result. The degree of the quotient will be one less than the original polynomial, so it will be a quadratic.
Use the quotient polynomial to find the remaining zeros of \( f(x) \) by solving the quadratic equation (either by factoring, completing the square, or using the quadratic formula). Remember to include \( x = -1 \) as a zero from the divisor.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
4m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form x - c. It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone. This method helps find the quotient and remainder efficiently.
Video consigliato:
05:10
Higher Powers of i

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by x - c, the remainder is equal to f(c). If the remainder is zero, then x = c is a root (zero) of the polynomial. This theorem helps verify if a candidate value is a zero of the polynomial.
Video consigliato:
05:10
Higher Powers of i

Finding Zeros of a Polynomial

Finding zeros involves solving f(x) = 0. After dividing the polynomial, the quotient can be factored or solved using other methods to find additional zeros. Identifying all zeros is essential for understanding the polynomial's behavior and graph.
Video consigliato:
03:42
Finding Zeros & Their Multiplicity
Pratica correlata
Domanda del libro di testo

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. 2x3−x2−9x−4=0

603
views
Domanda del libro di testo

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x3≥9x2x^3≥9x^2

528
views
Domanda del libro di testo

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x3≤4x2

508
views
Domanda del libro di testo

Find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x3/(3x2+1)

682
views
Domanda del libro di testo

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=5x2−5x

969
views
Domanda del libro di testo

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x−4)/(x+3) > 0

578
views