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Chapter 5: Problem 69

Divide the difference between \(4 x^{3}+x^{2}-2 x+7\) and \(3 x^{3}-2 x^{2}-7 x+4\) by \(x+1\)

Short Answer

Expert verified
The polynomial result after division is \(x^2 + 2x + 3\).

Step by step solution

01

Perform Polynomial Subtraction

Firstly, the difference between the two polynomials \(4x^3 + x^2 - 2x + 7\) and \(3x^3 - 2x^2 - 7x + 4\) should be calculated. This can be done by subtracting corresponding terms of the second polynomial from the first. Thus, the difference is \((4x^3 - 3x^3) + (x^2 - -2x^2) + (-2x - -7x) + (7 - 4) = x^3 + 3x^2 + 5x + 3\).
02

Perform Polynomial Division

Now, divide the obtained polynomial \(x^3 + 3x^2 + 5x + 3\) by the divisor \(x+1\). This can be done using polynomial long division. In first step, \(x^2\) times the divisor subtracted from \(x^3 + 3x^2\) leaves \(2x^2\); then, \(2x\) times the divisor is subtracted from \(2x^2 + 5x\), leaving \(3x\); finally, \(3\) times the divisor is subtracted from \(3x + 3\), leaving no remainder. Thus, the quotient is \(x^2 + 2x + 3\).
03

Representation of Final Answer

Putting it all together, the final solution for the given problem after performing the polynomial subtraction and division, is the quotient \(x^2 + 2x + 3\).

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Key Concepts

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

Polynomial Subtraction
Polynomial subtraction is the first step in this exercise. To subtract polynomials, we align both lists of terms and subtract corresponding terms. For example, subtract the polynomial \(3x^3 - 2x^2 - 7x + 4\) from \(4x^3 + x^2 - 2x + 7\). Align terms based on their degree:
  • Subtract the cubic terms: \(4x^3 - 3x^3 = x^3\)
  • Subtract the quadratic terms: \(x^2 - (-2x^2) = 3x^2\)
  • Subtract the linear terms: \(-2x - (-7x) = 5x\)
  • Subtract the constant terms: \(7 - 4 = 3\)
This yields the polynomial \(x^3 + 3x^2 + 5x + 3\). The main idea here is to handle each term individually, being mindful of signs, especially when subtracting.
Division of Polynomials
Once you obtain the polynomial \(x^3 + 3x^2 + 5x + 3\), it's time to divide it. Polynomial division is a technique used often to simplify expressions or solve polynomial equations. This process checks how many times a divisor can "fit" into another polynomial.
  • Start with the highest degree term of your polynomial and divide it by the highest degree term of the divisor.
  • Subtract the result from the original polynomial and repeat the process with the new leading term.
  • Continue until no terms are left or the remainder's degree is less than the divisor's degree.
In this problem, that divisor is \(x+1\). The result, known as a quotient, provides answers to the polynomial division: \(x^2 + 2x + 3\). The process helps us simplify complex polynomial expressions by systematically eliminating terms.
Polynomial Long Division
Polynomial long division is similar to numerical long division. Here's how it breaks down:Begin with the highest degree term of your dividend (the polynomial from subtraction), divided by the highest degree of the divisor.
  • Divide \(x^3\) by \(x\) to get \(x^2\).
  • Multiply \(x^2\) by \(x + 1\) (your divisor) and subtract the result from the original polynomial: \(x^3 + 3x^2 + 5x + 3\) minus \(x^3 + x^2\).
  • This leaves \(2x^2 + 5x + 3\).
  • Repeat the steps for \(2x^2\) and \(5x\) until you end up with zero remaining or a degree lower than your divisor.
This systematic method ensures no terms are missed, providing a clean solution to polynomial division problems. The given exercise results in a quotient of \(x^2 + 2x + 3\), eliminating the remainder perfectly.

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Most popular questions from this chapter

Express \(\frac{7}{8}\) as a decimal.

Explain the quotient rule for exponents. Use \(\frac{3^{6}}{3^{2}}\) in your explanation.

When dividing a binomial into a polynomial with missing terms, explain the advantage of writing the missing terms with zero coefficients.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Each statement applies to the division problem $$\frac{x^{3}+1}{x+1}$$ The purpose of writing \(x^{3}+1\) as \(x^{3}+0 x^{2}+0 x+1\) is to keep all like terms aligned.

In Exercises \(156-163\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$534.7=5.347 \times 10^{3}$$
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