Textbook Question
Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.
2
views
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- 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
Chapter 4, Problem 75
When 2x2−7x+9 is divided by a polynomial, the quotient is 2x-3 and the remainder is 3. Find the polynomial.
Verified step by step guidance1
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 \(D(x)\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
Guided course
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.
Recommended video:
Guided course
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.
Recommended video:
05:01Identifying Intervals of Unknown Behavior
Related Practice
Textbook Question
In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5
Textbook Question
Use synthetic division to show that 5 is a solution of x^4−4x^3−9x^2+16x+20=0. Then solve the polynomial equation.
6
views
Textbook Question
Solve the variation problems in Exercises 77–82. The distance that a body falls from rest is directly proportional to the square of the time of the fall. If skydivers fall 144 feet in 3 seconds, how far will they fall in 10 seconds?
1
views
Textbook Question
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x2+x−12)/(x2−4)
Textbook Question
Follow the seven steps to graph each rational function. f(x)=x4/(x2+2)