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Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.
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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 75
When 2x2−7x+9 is divided by a polynomial, the quotient is 2x-3 and the remainder is 3. Find the polynomial.
Guida verificata passo dopo passo1
Recall the polynomial division identity: Dividend = (Divisor) × (Quotient) + Remainder.
Let the divisor polynomial be denoted as \(D(x)\). According to the problem, the dividend is \(2x^{2} - 7x + 9\), the quotient is \(2x - 3\), and the remainder is 3.
Set up the equation: \(2x^{2} - 7x + 9 = D(x) \times (2x - 3) + 3\).
Rearrange the equation to isolate \(D(x)\): \(D(x) = \frac{2x^{2} - 7x + 9 - 3}{2x - 3} = \frac{2x^{2} - 7x + 6}{2x - 3}\).
Perform polynomial division or factorization on the numerator \(2x^{2} - 7x + 6\) to simplify the expression and find \(D(x)\).
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Polynomial Division
Polynomial division involves dividing one polynomial by another, resulting in a quotient and a remainder. The dividend equals the divisor multiplied by the quotient plus the remainder. Understanding this relationship is essential to reconstruct the divisor when the dividend, quotient, and remainder are known.
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05:13
Introduction to Polynomials
Forming Equations from Polynomial Expressions
Given the dividend, quotient, and remainder, you can set up an equation: Dividend = Divisor × Quotient + Remainder. This equation allows you to solve for the unknown polynomial divisor by expressing it in terms of the known polynomials and constants.
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05:16Standard Form of Polynomials
Solving for Unknown Polynomials
To find the unknown polynomial divisor, rearrange the equation and perform algebraic operations such as polynomial subtraction and division. Matching coefficients of corresponding powers of x helps determine the coefficients of the unknown polynomial.
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05:01Identifying Intervals of Unknown Behavior
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