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Chapter 11: Problem 14

Use synthetic division to divide. $$ \frac{4 x^{2}-5 x-20}{x-4} $$

Short Answer

Expert verified
The quotient is 4x + 11 with remainder 24.

Step by step solution

01

- Set Up the Synthetic Division

Write the coefficients of the polynomial: 4, -5, -20. Place the divisor's root (4) to the left side.
02

- Bring Down the Leading Coefficient

Bring down the first coefficient (4) directly below the line.
03

- Multiply and Add

Multiply 4 (the number just written) by 4 (the divisor's root) and write the result (16) underneath the next coefficient (-5). Add the numbers in this column to obtain the new value (11).
04

- Repeat the Process

Multiply the result from the previous column (11) by the divisor's root (4) and write the result (44) underneath the last coefficient (-20). Add these numbers to obtain the final value (24).
05

- Interpret the Results

The quotient of the division is written in terms of the new coefficients obtained: 4 and 11. Thus, the quotient is 4x + 11 with the remainder being 24.

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Key Concepts

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

Polynomial Division
Polynomial division is a process used to divide one polynomial by another. It is similar to the long division method used with numbers. Just like with numerical division, the goal is to find how many times the divisor polynomial can be subtracted from the dividend polynomial without going negative. This process helps in simplifying complex polynomial expressions. In the given exercise, synthetic division is a simplified form of polynomial division. It specifically applies when the divisor is a linear polynomial in the form of (x - c).
The Remainder Theorem
The Remainder Theorem is a helpful tool in polynomial division. It states that the remainder of the division of a polynomial P(x) by a linear divisor (x - c) is equal to P(c). In simpler terms, if you substitute c into the polynomial P(x), you will get the remainder that results from dividing P(x) by (x - c). For example, in our exercise, we divided 4x^2 - 5x - 20 by (x - 4). Using the Remainder Theorem, substituting 4 into the polynomial, 4(4)^2 - 5(4) - 20 gives 24. This confirms that the remainder is 24.
Algebraic Long Division
Algebraic long division is another method for dividing polynomials. It is a more general and detailed method than synthetic division and is applicable for any polynomial divisors. To perform algebraic long division:
  • Write the dividend and divisor in standard form.
  • Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
  • Multiply the whole divisor by this term and subtract the result from the dividend.
  • Repeat the process with the new dividend until the degree of the remainder is less than the degree of the divisor.

In synthetic division, we rely on a streamlined version that uses coefficients and the root of the divisor to achieve a faster result. It's especially useful because it's quicker and involves less notation.

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Most popular questions from this chapter

Graph each rational function. $$f(x)=\frac{(x-5)(x-2)}{x^{2}+9}$$

Use synthetic division to divide. $$ \frac{3 x^{2}+5 x-12}{x+3} $$

Graph each rational function. $$f(x)=\frac{-4 x}{3 x-1}$$

Approximate to the nearest hundredth the coordinates of the turning point in the given interval of the graph of each polynomial function. \(f(x)=x^{3}+4 x^{2}-8 x-8, \quad[-3.8,-3]\)

Use synthetic division to determine whether the given number is a zero of the polynomial function. $$ 3 ; \quad f(x)=2 x^{3}-6 x^{2}-9 x+27 $$
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