/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Two polynomials \(P\) and \(D\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 5

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=x^{4}-x^{3}+4 x+2, \quad D(x)=x^{2}+3\)

Short Answer

Expert verified
Quotient is \( x^2 - x - 3 \), remainder is \( 7x + 11 \).

Step by step solution

01

Verify the Degree of Polynomials

First, we need to confirm that the degree of the polynomial \( P(x) = x^4 - x^3 + 4x + 2 \) is greater than or equal to the degree of \( D(x) = x^2 + 3 \). The degree of \( P(x) \) is 4, and the degree of \( D(x) \) is 2, so division is appropriate here.
02

Set up Long Division

We begin by setting up our long division. The dividend is \( x^4 - x^3 + 0x^2 + 4x + 2 \) (note the placeholder for \( x^2 \)), and the divisor is \( x^2 + 3 \). Our goal is to determine how many times the leading term of the divisor can be multiplied to yield the leading term of the dividend.
03

Divide Leading Terms

Divide the leading term of the dividend, \( x^4 \), by the leading term of \( x^2 + 3 \), which is \( x^2 \). This gives \( x^2 \) as the first term of the quotient \( Q(x) \).
04

Multiply and Subtract

Multiply the entire divisor \( x^2 + 3 \) by the term \( x^2 \) to get \( x^4 + 3x^2 \). Subtract this from the original polynomial: \\( (x^4 - x^3 + 0x^2 + 4x + 2) - (x^4 + 3x^2) = -x^3 - 3x^2 + 4x + 2 \).
05

Repeat Division Process

Next, divide \( -x^3 \) by \( x^2 \) to get \( -x \). This becomes the next term of the quotient. Multiply \( -x \) by \( x^2 + 3 \) to get \( -x^3 - 3x \), and subtract this from what remains of the dividend: \\(-x^3 - 3x^2 + 4x + 2 - (-x^3 - 3x) = -3x^2 + 7x + 2 \).
06

Final Division Step

Divide \( -3x^2 \) by \( x^2 \) to get \( -3 \). Multiply \( -3 \) by \( x^2 + 3 \) to obtain \( -3x^2 - 9 \). Subtract from the remaining polynomial: \\(-3x^2 + 7x + 2 - (-3x^2 - 9) = 7x + 11 \).
07

Express Final Quotient and Remainder

The division process is complete when the degree of the remainder is less than the degree of the divisor. Our quotient \( Q(x) \) is \( x^2 - x - 3 \) and the remainder \( R(x) \) is \( 7x + 11 \). Thus, \( P(x) = (x^2 + 3)(x^2 - x - 3) + (7x + 11) \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Long Division
In polynomial math, long division is similar to the long division you use with numbers. It's a systematic process to divide one polynomial by another, resulting in a quotient and a remainder.
To carry out long division, you must carefully line up terms by degree, just like organizing numbers in math class.
  • The first step is dividing the leading term of the dividend by the leading term of the divisor. This gives you the first term of your quotient.
  • Next, you multiply the entire divisor by this term, and concurrently, subtract the result from the original polynomial.
  • Repeat this process with the new polynomial remainder until the degree of the remainder is less than the divisor.
It's crucial to insert any missing terms with zeros as placeholders, as seen with the polynomial \(x^4 - x^3 + 0x^2 + 4x + 2\). This ensures all coefficients line up correctly under subtraction.
Synthetic Division
Synthetic division serves as a shortcut method to divide polynomials, particularly when the divisor is a first-degree binomial like \(x - c\). Instead of handling the entire polynomial like in long division, you work with coefficients only.
Here's the streamlined process:
  • Arrange the polynomial's coefficients in decreasing order, filling any gaps with zeros.
  • Write the zero of the divisor (i.e., opposite of the constant term in \(x-c\) ) at the left.
  • Bring down the leading coefficient to start your new row, then multiply and add successively using the zero.
  • The final row gives the quotient coefficients, and the last number is the remainder.
Synthetic division simplifies the task when it's applicable, providing a faster way to find the quotient and remainder compared to long division.
Degree of Polynomials
The degree of a polynomial is the highest power of the variable in its expression, playing a crucial role in division.
When dividing, always ensure the degree of the dividend is at least as large as the degree of the divisor. This is because:
  • Division is only meaningful when the dividend's degree is greater than or equal to the divisor's.
  • The quotient will reflect a drop in degree by the degree of the divisor.
  • The remainder will have a degree less than the divisor.
For instance, dividing \(P(x) = x^4 - x^3 + 4x + 2\) by \(D(x) = x^2 + 3\) involves degrees 4 and 2, respectively, making the operation viable. Attention to degrees ensures you align your division steps correctly.
Remainder Theorem
The Remainder Theorem is a handy concept when you're working with polynomial division, indicating that the remainder you've calculated has a specific meaning. It states if you divide a polynomial \(P(x)\) by a linear factor \(x-c\), the remainder is \(P(c)\).
This theorem allows for quick checks:
  • If you substitute \(c\) into the polynomial \(P(x)\), the result is exactly the remainder.
  • It helps verify calculations by evaluating the original polynomial at \(c\), without going through every step of division again.
  • It even aids in determining possible zeroes for polynomials.
For our exercise, since \(D(x)\) is not linear, the Remainder Theorem doesn't directly apply, but understanding its implications is important when encountering linear divisors.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

How Many Real Zeros Can a Polynomial Have? Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial. (a) A polynomial of degree 3 that has no real zeros (b) A polynomial of degree 4 that has no real zeros (c) A polynomial of degree 3 that has three real zeros, only one of which is rational (d) A polynomial of degree 4 that has four real zeros, none of which is rational What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes鈥� Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6 $$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes鈥� Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=2 x^{4}+3 x^{3}-4 x^{2}-3 x+2 $$

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ \begin{array}{l}{U \text { has degree } 5, \text { zeros } \frac{1}{2},-1, \text { and }-i, \text { and leading coefficient }} \\ {4 ; \text { the zero }-1 \text { has multiplicity } 2 .}\end{array} $$

The Depressed Cubic The most general cubic (third-degree) equation with rational coefficients can be written as $$x^{3}+a x^{2}+b x+c=0$$ (a) Show that if we replace \(x\) by \(X-a / 3\) and simplify, we end up with an equation that doesn't have an \(X^{2}\) term, that is, an equation of the form $$ X^{3}+p X+q=0 $$ This is called a depressed cubic, because we have depressed the quadratic term. (b) Use the procedure described in part (a) to depress the equation \(x^{3}+6 x^{2}+9 x+4=0\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.