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. (x+1)(x+2)(x+3)≥0
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 4, Problema 31
Divide using synthetic division. (2x5−3x4+x3−x2+2x−1)/(x+2)
Guida verificata passo dopo passo1
Identify the divisor and rewrite it in the form \( x - c \). Since the divisor is \( x + 2 \), rewrite it as \( x - (-2) \), so \( c = -2 \).
Write down the coefficients of the dividend polynomial \( 2x^{5} - 3x^{4} + x^{3} - x^{2} + 2x - 1 \). The coefficients are: \( 2, -3, 1, -1, 2, -1 \).
Set up the synthetic division by writing \( c = -2 \) to the left and the coefficients in a row to the right.
Bring down the first coefficient (2) as it is. Then multiply it by \( c = -2 \) and write the result under the next coefficient. Add the column and write the sum below. Repeat this multiply-and-add process for all coefficients.
The numbers obtained at the bottom row (except the last one) are the coefficients of the quotient polynomial, starting from one degree less than the original polynomial. The last number is the remainder.
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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.
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Higher Powers of i
Polynomial Coefficients and Degree
Understanding the coefficients and degree of a polynomial is essential for synthetic division. The degree indicates the highest power of x, and coefficients are the numerical factors of each term, which are arranged in descending order of degree for the division process.
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Division by a Binomial of the Form x + c
When dividing by a binomial like x + 2, it is rewritten as x - (-2) to apply synthetic division. Recognizing this form allows you to correctly identify the value of c (here, -2) used in the synthetic division steps.
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