Problem 28
Question
Use synthetic division to find the quotient and remainder If the first polynomial is divided by the second. $$9 x^{3}-6 x^{2}+3 x-4 ; \quad x-\frac{1}{3}$$
Step-by-Step Solution
Verified Answer
Quotient: \(9x^2 - 3x + 2\); Remainder: \(-\frac{10}{3}\).
1Step 1: Set up the synthetic division
Identify the divisor, which is in the form \(x - c\). Here, \(c = \frac{1}{3}\). Write down the coefficients of the dividend polynomial \(9x^{3} - 6x^{2} + 3x - 4\), which are \(9\), \(-6\), \(3\), and \(-4\). These coefficients are placed in a row for synthetic division.
2Step 2: Begin the synthetic division process
Write \(\frac{1}{3}\) to the left of the coefficients of the polynomial. Bring down the leading coefficient \(9\) as it is.
3Step 3: Multiply and add the first coefficient
Multiply \(9\) by \(\frac{1}{3}\) to get \(3\). Add this result to the next coefficient, \(-6\), resulting in \(-3\). Write \(-3\) below.
4Step 4: Continue the process with the second coefficient
Multiply \(-3\) by \(\frac{1}{3}\) to get \(-1\). Add this result to the next coefficient, \(3\), resulting in \(2\). Write \(2\) below.
5Step 5: Multiply and add the last coefficient
Multiply \(2\) by \(\frac{1}{3}\) to get \(\frac{2}{3}\). Add this result to the last coefficient, \(-4\), resulting in \(-\frac{10}{3}\). Write \(-\frac{10}{3}\), which is the remainder.
6Step 6: Interpret the result
The quotient is given by the numbers \(9, -3, 2\) and represents \(9x^2 - 3x + 2\). The remainder of this division is \(-\frac{10}{3}\). Thus, \(9x^3 - 6x^2 + 3x - 4 = (x - \frac{1}{3})(9x^2 - 3x + 2) + (-\frac{10}{3})\).
Key Concepts
Polynomial DivisionRemainder TheoremPolynomial Coefficients
Polynomial Division
Polynomial division is much like long division, but it is used for dividing polynomials. When you have a polynomial (the dividend) and you want to divide it by another polynomial (the divisor), you're essentially breaking it down to find out how many times the divisor fits into the dividend.
In this exercise, we are dividing a third-degree polynomial by a first-degree polynomial. The goal here is to obtain both a quotient and a remainder. The quotient is the polynomial that represents how many times the divisor fits into the dividend, and the remainder is what's left over.
Synthetic division is a streamlined technique used particularly when the divisor is of the form \(x - c\), where \(c\) is a constant. This makes the process faster and simpler than traditional long division. It especially comes in handy when dealing with polynomials in algebraic expressions.
In this exercise, we are dividing a third-degree polynomial by a first-degree polynomial. The goal here is to obtain both a quotient and a remainder. The quotient is the polynomial that represents how many times the divisor fits into the dividend, and the remainder is what's left over.
Synthetic division is a streamlined technique used particularly when the divisor is of the form \(x - c\), where \(c\) is a constant. This makes the process faster and simpler than traditional long division. It especially comes in handy when dealing with polynomials in algebraic expressions.
Remainder Theorem
The remainder theorem is a valuable tool in algebra that helps simplify polynomial division. According to this theorem, if a polynomial \(f(x)\) is divided by \(x - c\), the remainder of this division is equal to \(f(c)\).
This means if you substitute \(c\) into the polynomial and evaluate it, you will get the remainder without having to go through the entire division process. This is great for checking the accuracy of your division. In our example, if you substitute \(\frac{1}{3}\) into the polynomial \(9x^{3} - 6x^{2} + 3x - 4\), computing it gives you the same remainder \(-\frac{10}{3}\), confirming the result of our synthetic division.
Using the remainder theorem not only verifies results but also provides insight into the relationship between polynomial roots and remainders.
This means if you substitute \(c\) into the polynomial and evaluate it, you will get the remainder without having to go through the entire division process. This is great for checking the accuracy of your division. In our example, if you substitute \(\frac{1}{3}\) into the polynomial \(9x^{3} - 6x^{2} + 3x - 4\), computing it gives you the same remainder \(-\frac{10}{3}\), confirming the result of our synthetic division.
Using the remainder theorem not only verifies results but also provides insight into the relationship between polynomial roots and remainders.
Polynomial Coefficients
Polynomial coefficients are the numbers that multiply the variables or powers of the variables in a polynomial. Understanding them is crucial when setting up and performing synthetic division.
Consider the polynomial \(9x^{3} - 6x^{2} + 3x - 4\). Here, the coefficients are \(9\), \(-6\), \(3\), and \(-4\), respectively.
Consider the polynomial \(9x^{3} - 6x^{2} + 3x - 4\). Here, the coefficients are \(9\), \(-6\), \(3\), and \(-4\), respectively.
- The leading coefficient is the number in front of the highest degree term, so for our polynomial, it's \(9\).
- These coefficients are essential inputs into the synthetic division process, lining up to perform the calculations needed to find the quotient and remainder.
- The arrangement and treatment of coefficients directly affect the outcome of synthetic division.
Other exercises in this chapter
Problem 28
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Does there exist a polynomial of degree 3 with real coefficients that has zeros \(1,-1,\) and \(i\) ? Justify your answer.
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Density at a point \(A\) thin flat plate is situated in an \(x y\) -plane such that the density \(d\) (in Ib/ft \(^{2}\) ) at the point \(P(x, y)\) is inversely
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