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Chapter 4: Problem 26

Find the quotient and remainder using synthetic division. \(\frac{4 x^{2}-3}{x+5}\)

Short Answer

Expert verified
Quotient: \(4x - 20\), Remainder: 97.

Step by step solution

01

Setup for synthetic division

First, identify the coefficients of the polynomial dividend, which is \(4x^2 - 3\). Since it is a quadratic polynomial, the coefficients are [4, 0, -3]. Note that the middle term \(x\) is missing, so we include it as 0. The divisor is \(x + 5\), and for synthetic division, we set \(x + 5 = 0\) to find the value used, which is \(-5\). Write these across the synthetic division setup.
02

Perform synthetic division

Using the value \(-5\) from the divisor, start the synthetic division process. Bring down the leading coefficient 4 to the bottom row. Multiply it by \(-5\) (the value from the divisor) and write the result under the next coefficient, which is 0. Add down: \(0 - 20 = -20\). Continue by multiplying \(-20\) by \(-5\), resulting in 100. Add \(-3 + 100 = 97\).
03

Interpret the results

From the synthetic division, the bottom row with results is \([4, -20, 97]\). This corresponds to the quotient and remainder of the division. The quotient is one degree lower than the dividend polynomial (which was \(4x^2\)), so the quotient here is \(4x - 20\). The remainder is the last number, which is 97.
04

Write the final answer

Combine the terms from the previous steps. The quotient of the division is \(4x - 20\), and the remainder is 97. Thus, the expression can be rewritten as: \(4x - 20 + \frac{97}{x+5}\).

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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 method used to divide polynomials, much like how we divide numbers. It helps in simplifying expressions and finding factors of polynomials. There are two main types of polynomial division: long division and synthetic division.
  • Long division is similar to numeric division, involving writing down terms, subtracting, and bringing down the next term.
  • Synthetic division, on the other hand, is a shortcut method specifically used when dividing by a linear divisor of the form \(x + c\), where \(c\) is a constant.
This method requires less writing and is generally quicker.
However, it can only be applied under certain conditions, specifically when the divisor is a first-degree polynomial.
Quotient and Remainder
In any division operation, when you divide one number by another, you get a quotient and sometimes a remainder. When we apply this concept to polynomials, the operation is similar. After performing polynomial division with a polynomial dividend and a linear divisor, you get:
  • The quotient: A polynomial one degree lower than the original dividend polynomial.
  • The remainder: A constant or polynomial with a degree lower than the divisor.
In the example given, the polynomial \(4x^2 - 3\) was divided by \(x + 5\), resulting in a quotient of \(4x - 20\) and a remainder of 97.
The expression can thus be rewritten to show both, the quotient and remainder, as \(4x - 20 + \frac{97}{x+5}\).
Remember, if the remainder is zero, it implies that the divisor is a factor of the dividend.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. A polynomial is a special type of algebraic expression where variables are raised to whole number exponents and are combined using these operations.
  • Polynomials can be classified based on the number of terms they have, such as monomials (one term), binomials (two terms), and trinomials (three terms).
  • The degree of the polynomial is determined by the highest exponent of the variable.
In the division exercise, the polynomial \(4x^2 - 3\) is a quadratic polynomial because the highest degree term is \(x^2\).
Understanding how to manipulate these expressions is crucial for solving equations, factoring, and simplifying algebraic fractions.
Using synthetic division with these expressions allows us to efficiently find the relationship between a dividend, its quotient, and the remainder.

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Most popular questions from this chapter

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=3 x^{3}-5 x^{2}-8 x-2 $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=4 x^{4}+4 x^{3}+5 x^{2}+4 x+1 $$

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ Q \text { has degree } 3, \text { and zeros } 0 \text { and } i $$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=2 x^{4}+15 x^{3}+17 x^{2}+3 x-1 $$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=2 x^{3}+5 x^{2}+x-2 ; \quad a=-3, b=1 $$
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