/*! 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 88 Factor by grouping, if possible,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 88

Factor by grouping, if possible, and check. $$ y^{4}-y^{3}-y+y^{2} $$

Short Answer

Expert verified
The factored form is y(y - 1)(y^2 + 1).

Step by step solution

01

- Group the terms

Group the polynomial's terms into two pairs: y^{4} - y^{3} - y + y^{2}
02

- Factor out the greatest common factor (GCF) from each pair

For the first pair (y^4 - y^3), factor out the GCF which is y^3: y^3(y - 1) For the second pair (-y + y^2), factor out the GCF which is y: y(y - 1)
03

- Factor out the common binomial factor

Now, we notice that (y - 1) is common in both terms:y^3(y - 1) + y(y - 1) can be factored as: (y - 1)(y^3 + y)
04

- Simplify further if possible

Notice that (y^3 + y) can be factored further by factoring out y: y(y^2 + 1) Thus, the fully factored form is:y(y - 1)(y^2 + 1)
05

- Verify the factored form by multiplying it out

Expand the factored form to check if it matches the original polynomial: y(y - 1)(y^2 + 1) = y(y(y^2 + 1) - (y^2 + 1)) = y(y^3 + y - y^2 - 1) = y^4 - y^3 + y^2 - y which is equivalent to the original polynomial, so our factorization is correct.

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.

Greatest Common Factor (GCF)
To start factoring by grouping, the Greatest Common Factor (GCF) is essential. The GCF is the largest factor that divides two or more terms. For example, consider the terms in the polynomial given: y^4 - y^3 - y + y^2To find the GCF of any group of terms, look for the highest power of common variables and constants.
In y^4 - y^3the GCF is y^3, because it's the highest power of y that appears in both terms.
Similarly, in y^2 and y, the GCF is y. Identifying the GCF simplifies terms and brings out common factors, making it easier to group and factor further.
Polynomial Factoring
Polynomial factoring is about breaking down a polynomial into simpler factors that, when multiplied, yield the original polynomial. In this exercise, the polynomial y^4 - y^3 - y + y^2 is factored by grouping.
We first group the terms: y^4 - y^3 and -y + y^2.
Next, we factor out the GCF from each group:
  • From y^4 - y^3, we factor out y^3: y^3(y - 1)
  • From -y + y^2, we factor out y: y(y - 1)

Then, we notice that (y - 1) is a common binomial factor. This common factor can be factored out to give: (y - 1)(y^3 + y).
We can simplify further by factoring y out of y^3 + y to get: y(y^2 + 1).
The fully factored form of the polynomial is then: y(y - 1)(y^2 + 1).
Algebraic Verification
After factoring, it is crucial to verify our work by expanding the factors to see if they give us the original polynomial. We start with the fully factored form: y(y - 1)(y^2 + 1).
First, expand y(y^2 + 1):
  • y(y^2 + 1) = y^3 + y

This gives us: (y - 1)(y^3 + y).
Next, distribute (y - 1):
  • (y - 1)(y^3 + y) = y^3(y - 1) + y(y - 1)

Expanding the terms: y^3(y - 1) + y(y - 1) = y^4 - y^3 + y^2 - y.
This matches the original polynomial y^4 - y^3 - y + y^2. Since our expanded form equals the original polynomial, our factorization is correct. Verification like this ensures the factorization process was accurate and complete.

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

Solve. Round any irrational solutions to the nearest thousandth. $$ 27 x^{3}+18 x^{2}=12 x+8 $$

Write an equivalent expression by factoring. Assume that all exponents are natural numbers. $$ 3 a^{n+1}+6 a^{n}-15 a^{n+2} $$

Find the zeros of each function. $$ f(x)=x^{2}-16 $$

Factor completely. Assume that variables in exponents represent positive integers. $$ 0.09 x^{8}+0.48 x^{4}+0.64 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ x^{2}+2 x y+y^{2}-9 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

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.