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Chapter 5: Problem 120

Factor each polynomial completely. $$a b-a y+3 b-3 y$$

Short Answer

Expert verified
(a + 3)(b - y)

Step by step solution

01

Group terms

Group the terms in pairs to make it easier to factor by grouping. Group a b - a yand 3 b - 3 y.
02

Factor out common terms from each group

Factor out the common factor from each group: a (b - y) from the first group and 3 (b - y) from the second group.
03

Factor out the common binomial factor

Both terms now have a common binomial factor (b - y). Factor this out to get:(a + 3)(b - y).

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Key Concepts

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

Polynomial Factorization
Polynomial factorization is a technique used to break down a polynomial into simpler components called factors. Factoring polynomials can come in handy for solving equations or simplifying expressions.
The main idea is to express the polynomial as a product of its factors. For example, the polynomial given in the exercise is \(a b - a y + 3 b - 3 y\).
To factorize it, we look for ways to group and simplify the terms, as detailed in the solution.
This method helps you transform complex polynomial expressions into a form that's easier to work with. Once you master factoring, solving polynomial equations will become much simpler.
Common Factors
Common factors are the numbers or terms that are shared by the terms in an expression. Identifying common factors is often the first step in the factoring process.
In the exercise, we grouped terms to find common factors within those groups.
Let's take the polynomial \(a b - a y + 3 b - 3 y\):
Group \(a b - a y\) and \(3 b - 3 y\).
For the first group \(a b - a y\), we can factor out an 'a', resulting in \(a (b - y)\).
For the second group \(3 b - 3 y\), we can factor out a '3', resulting in \(3 (b - y)\).
Always check for common factors before attempting to use other factoring methods.
Binomial Factors
A binomial is a polynomial with two terms separated by either a plus or minus sign. When factoring, we often look for common binomial factors to simplify the expression.
In our example, after factoring out the common terms, we get \(a (b - y) + 3 (b - y)\).
Notice that \((b - y)\) is a common binomial factor in both terms.
Therefore, we can factor \((b - y)\) out, resulting in \( (a + 3)(b - y) \).
This technique allows us to factor a polynomial by recognizing and extracting binomial factors, which makes the expression simpler and more manageable. Understanding how to identify and factor out binomial factors is crucial for mastering polynomial factorization.

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Most popular questions from this chapter

Factor each polynomial using the trial-and-error method. $$ 3 x^{2}+26 x-9 $$

Firing a slingshot. A girl uses a slingshot to propel a stone upward at 64 feet per second from a window that is 80 feet above the ground. The height of the stone above the ground (in feet) at time \(t\) (in seconds) is given by $$h(t)=-16 t^{2}+64 t+80$$ Find the time at which the stone strikes the ground.

If you divide each side of \(x^{2}=x\) by \(x,\) you get \(x=1\) If you subtract \(x\) from each side of \(x^{2}=x,\) you get \(x^{2}-x=0,\) which has two solutions. Which method is correct? Explain.

Factor each polynomial using the trial-and-error method. $$ y^{2}-23 y+130 $$

Factor each polynomial by grouping. $$ 2 q h-h+8 q-4 $$
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