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Chapter 5: Problem 90

Factor by grouping. \(8 s^{2}+6 s y-4 s t-3 y t\)

Short Answer

Expert verified
(4s + 3y)(2s - t)

Step by step solution

01

- Group the terms

Group the terms in pairs. Group the first two terms together and the last two terms together: 8s^{2} + 6sy - 4st - 3yt.
02

- Factor out the Greatest Common Factor (GCF) from each group

Factor out the GCF from each group. For the first group (8s^{2} + 6sy), the GCF is 2s, and for the second group (-4st - 3yt), the GCF is -t: 2s(4s + 3y) - t(4s + 3y).
03

- Factor out the common binomial factor

Notice that each group contains the common binomial factor (4s + 3y). Factor this out from the expression: (4s + 3y)(2s - t).

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Key Concepts

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

polynomial factoring
Polynomial factoring is the process of breaking down a complex polynomial into simpler polynomials that, when multiplied together, give the original polynomial. This process is crucial in solving polynomial equations, simplifying expressions, and analyzing polynomial functions.

If you're dealing with a polynomial and want to factor it, you often look for patterns or use techniques like grouping to break it into pieces. For example, with the expression in the exercise \(8s^{2} + 6sy - 4st - 3yt\), our goal is to split it into products of simpler terms.

When polynomials are factored successfully, it makes equations easier to solve. Always make sure you simplify as much as possible.
greatest common factor
The greatest common factor (GCF) is an important concept in polynomial factoring. It's the largest factor that can be evenly divided into each term of a polynomial. This helps when you're grouping terms to simplify the polynomial.

Take a look at the polynomial we have, \(8s^{2} + 6sy - 4st - 3yt\). In Step 1 from the solution, we grouped the terms into pairs: \(8s^{2} + 6sy\) and \(-4st - 3yt\).

To simplify each group, we find the GCF:
  • For \(8s^{2} + 6sy\), the GCF is \(2s\).
  • For \(-4st - 3yt\), the GCF is \(-t\).
Therefore, by factoring out the GCF, we simplify each group to: \(2s(4s + 3y)\) and \(-t(4s + 3y)\).

This dramatically simplifies our polynomial.
common binomial factor
A common binomial factor is a binomial that is a factor of each term in a polynomial. It's a crucial step in the factoring by grouping method, as it helps pull the polynomial apart neatly.

After factoring out the greatest common factors from the grouped terms, we have \(2s(4s + 3y) - t(4s + 3y)\). Both terms contain the binomial \((4s + 3y)\). This is our common binomial factor.

We can now factor out \((4s + 3y)\) from the expression:
  • Our expression becomes: \((4s + 3y)(2s - t)\).
Now, the polynomial \(8s^{2} + 6sy - 4st - 3yt\) is fully factored as \((4s + 3y)(2s - t)\). This method simplifies the polynomial significantly by using common binomial factors.

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Most popular questions from this chapter

Solve each problem. The product of the first and third of three consecutive integers is 3 more than 3 times the second integer. Find the integers.

Solve each problem. Find three consecutive odd integers such that the sum of all three is 42 less than the product of the second and third integers.

Factor each binomial completely. \(2 x^{3}-16 y^{3}\)

Factor each binomial completely. \(27 r^{3}+1000 s^{3}\)

Solve each problem. The product of two consecutive integers is 11 more than their sum. Find the integers.
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