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Chapter 2: Problem 29

Use synthetic division to divide. $$\left(6 x^{3}+7 x^{2}-x+26\right) \div(x-3)$$

Short Answer

Expert verified
The polynomial \(6x^3+7x^2-x+26\) divided by \(x-3\) using synthetic division is \(6x^2+39x+116\) with a remainder of 368.

Step by step solution

01

Arrange the coefficients

The first step is to arrange the coefficients of the polynomial in the highest power order. In this case, the coefficients are 6, 7, -1, and 26. Also, the value of x in \(x-3\) is 3.
02

Synthetic division set-up

Next, set up the synthetic division. Draw a line and write the coefficients (6, 7, -1, and 26) above the line. Beneath the line on the left side, write -3, the opposite value of which x is equal to in your divisor, \(x-3\).
03

Begin synthetic division - carry down step

Begin the synthetic division process by carrying down the first coefficient, which is 6.
04

Multiply and add

Multiply the value you brought down (6) by 3 (since x=3) and write the result under the next coefficient (7). Add these two values together and write the result under the line. Repeat the process with the obtained result until you reach the last coefficient. You should get results of 39, 116 and 368 respectively.
05

Interpret the Result

The result of synthetic division should then be interpreted. Starting from the right, the final number is the remainder and the polynomial degree is decreased by 1 each step to the left. So, the final polynomial is \(6x^2+39x+116\) with a remainder of 368.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficients in Polynomials
Polynomials are constructed of terms, where each term has a coefficient and a variable raised to a power. In \(6x^3 + 7x^2 - x + 26\), 6, 7, and -1 are the coefficients while 26 is a constant term. Identifying these correctly is crucial for synthetic division.
Remainder Theorem
The remainder theorem is a fundamental aspect of polynomial arithmetic. It states that when a polynomial \(f(x)\) is divided by a linear divisor of the form \(x - c\), the remainder is the value obtained by substituting the divisor's root \(c\) back into the polynomial. For the exercise \(6x^3 + 7x^2 - x + 26\) divided by \(x - 3\), according to the remainder theorem, by just plugging in \(x = 3\) into the polynomial, we will get the remainder without doing the full synthetic division process.

Therefore, the remainder theorem serves as a quick check on the final term obtained from synthetic division, ensuring the division was performed correctly. Additionally, this theorem is particularly useful in verifying the factors of a polynomial and is closely related to polynomial roots.

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Most popular questions from this chapter

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{3}-4 x^{2}+1\) (a) Upper: \(x=4\) (b) Lower: \(x=-1\)

Write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. Vertex: (2,3)\(;\) point: (0,2)

The total revenue \(R\) earned (in thousands of dollars) from manufacturing handheld video games is given by $$R(p)=-25 p^{2}+1200 p$$ where \(p\) is the price per unit (in dollars). (a) Find the revenues when the prices per unit are \(\$ 20\) \(\$ 25,\) and \(\$ 30\) (b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results.

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-x$$

Use a graphing utility to graph the quadratic function. Find the \(x\) -intercept(s) of the graph and compare them with the solutions of the corresponding quadratic equation when \(f(x)=0\) $$f(x)=2 x^{2}-7 x-30$$
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