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Divide using synthetic division. (x7+x5−10x3+12)/(x+2)
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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 29
Divide using synthetic division. (x4−256)/(x−4)
Guida verificata passo dopo passo1
Identify the divisor and the dividend. Here, the divisor is \(x - 4\), and the dividend is \(x^{4} - 256\).
Set up the synthetic division by writing the zero of the divisor \(x - 4\), which is \(4\), to the left.
Write the coefficients of the dividend polynomial \(x^{4} - 256\). Since some terms are missing, include zeros for those terms: coefficients are \(1\) (for \(x^{4}\)), \(0\) (for \(x^{3}\)), \(0\) (for \(x^{2}\)), \(0\) (for \(x\)), and \(-256\) (constant term).
Perform synthetic division: bring down the first coefficient, multiply it by \(4\), add to the next coefficient, and repeat this process across all coefficients.
Write the quotient polynomial using the results from synthetic division, noting that the degree of the quotient is one less than the dividend, and identify any remainder if present.
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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 and performing arithmetic operations in a tabular form.
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Higher Powers of i
Polynomial Coefficients
When using synthetic division, it is essential to identify and list the coefficients of the polynomial in descending order of degree. Missing terms must be represented by zero coefficients to maintain the correct alignment during the division process.
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Remainder and Quotient Interpretation
The result of synthetic division includes a quotient polynomial and possibly a remainder. The quotient has a degree one less than the original polynomial, and the remainder is a constant. Understanding how to interpret these results is key to completing the division.
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3:49Product, Quotient, and Power Rules of Logs
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