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Chapter 5: Problem 40

Factor each polynomial by grouping. $$2 a+2 b+w a+w b$$

Short Answer

Expert verified
The factored form is \((a + b)(2 + w)\).

Step by step solution

01

- Group the terms

Group the terms in pairs that have common factors: \( (2a + 2b) + (wa + wb) \)
02

- Factor out the common factor in each group

In the first group, factor out the common factor of 2. In the second group, factor out the common factor of w: \( 2(a + b) + w(a + b) \)
03

- Factor out the common binomial

Now, factor out the common binomial factor \( (a + b) \) from both terms: \( (a + b)(2 + w) \)

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Key Concepts

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

Grouping Method
The grouping method is a commonly used technique in algebra to factor polynomials. It involves arranging terms in pairs to identify common factors easily.
For example, in the polynomial \(2a + 2b + wa + wb\), we can use the grouping method to simplify it.
Here鈥檚 how you can apply this technique:

  • Step 1: Group the terms that have common factors. In this case, group \((2a + 2b)\) and \((wa + wb)\).

  • Step 2: Identify and factor out the greatest common factor (GCF) in each group. Factor out 2 from the first group and w from the second group to get \(2(a + b)\) and \(w(a + b)\).

  • Step 3: Combine the expressions by factoring out the common binomial \((a + b)\). This results in \((a + b)(2 + w)\).

The grouping method simplifies the polynomial factoring process by breaking it down into smaller, more manageable steps.
Common Factors
The concept of common factors is fundamental in the grouping method. A common factor is a term that divides each term in a group without leaving a remainder.

Let's revisit our example \(2a + 2b + wa + wb\).
After grouping, we have \((2a + 2b)\) and \((wa + wb)\).
In the first group, \(2\) is a common factor because it divides both \(2a\) and \(2b\).
In the second group, \(w\) is the common factor as it divides both \(wa\) and \(wb\).

  • Identifying Common Factors: Carefully scrutinize each group to find the term or number that is present in all terms of that group. In \((2a + 2b)\), the common factor is \(2\).

  • Factoring Out the Common Factor: Simply divide each term by the common factor and write it as a product. For the first group, \(2a + 2b\) becomes \(2(a + b)\).

Factoring out common factors simplifies the polynomial, making it easier to recognize common binomials and further simplify the expression.
Binomial Factors
A binomial factor in a polynomial is an expression that contains two terms. In our example, the binomial factor is \(a + b\).
After factoring out the common factors using the grouping method, we bring out the binomial factor next.

Here are a few steps to understand binomial factors better:
  • Step 1: After grouping, identify if the resulting expressions share a common binomial factor. In \(2(a + b) + w(a + b)\), the common binomial factor is \(a + b\).

  • Step 2: Factor the common binomial out as you would with a common numerical factor. So, \(2(a + b) + w(a + b)\) becomes \((a + b)(2 + w)\).

Understanding binomial factors helps in combining like terms and simplifying complex polynomials, enhancing your ability to manipulate algebraic expressions efficiently.

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Most popular questions from this chapter

Tossing a ball. A boy tosses a ball upward at 32 feet per second from a window that is 48 feet above the ground. The height of the ball above the ground (in feet) at time \(t\) (in seconds) is given by $$h(t)=-16 t^{2}+32 t+48$$ Find the time at which the ball strikes the ground.

Solve each equation for \(y .\) Assume a and b are positive numbers. $$a^{2} y^{2}-b^{2}=0$$

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.

Factor each polynomial using the trial-and-error method. $$ z^{2}-2 z-35 $$

Factor each polynomial using the trial-and-error method. $$ x^{2}-8 x+12 $$
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