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Chapter 0: Problem 34

Multiply and simplify. $$ (w+7)(6 w-3) $$

Short Answer

Expert verified
Combine terms to get: \[6w^2 + 39w - 21\].

Step by step solution

01

Apply the Distributive Property

Start by applying the distributive property, which states that \((a + b)(c + d) = a(c + d) + b(c + d)\). Here, write \((w + 7)\) and multiply it by each term in the expression \((6w - 3)\). This gives: \((w + 7)(6w) + (w + 7)(-3)\).
02

Multiply the Terms

Distribute \(w\) and \(7\) across \(6w - 3\): \[(w \times 6w + w \times -3) + (7 \times 6w + 7 \times -3)\]. This simplifies to: \[6w^2 - 3w + 42w - 21\].
03

Combine Like Terms

Combine the like terms \-3w\ and \42w\: \[6w^2 + 39w - 21\].

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Key Concepts

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

Distributive Property
The distributive property allows us to multiply a sum by a number by distributing the multiplication over each term in the sum. It's like breaking down a big problem into smaller parts.

In our exercise, we have \( (w + 7)(6w - 3) \). We can apply the distributive property by multiplying \( (w+7) \) to each term in \( (6w - 3) \). This is expressed as:

\[(w + 7)(6w - 3) = (w + 7) \times 6w + (w + 7) \times (-3)\].

This step sets the stage for simplifying the expression into more manageable parts. It's like laying the foundation before building a house.
Combining Like Terms
Once we have applied the distributive property, we get several terms that need to be simplified further. First, let's break down the multiplication:

\[6w^2 - 3w + 42w - 21\]

Here, \(6w^2 \) stands alone as it has no similar term. However, \(-3w \) and \(+ 42w \) are 'like terms' because they both have the variable \(w \). Like terms can be combined by simply doing the arithmetic on their coefficients, allowing us to simplify:

\[-3w + 42w = 39w\]

So now our expression looks like this:

\[6w^2 + 39w - 21\]

Combining like terms makes the expression simpler, easing further steps.
Simplification of Expressions
Simplification is the process of making an expression easier to read or solve by performing all possible arithmetic operations and combining like terms.

In our original problem, we started by distributing, multiplied the terms, and then combined like terms. Now we have:

\[6w^2 + 39w - 21\]

This is the simplified form of our original expression. There aren鈥檛 any further simplifications we can make. Our final expression is neat and easy to understand! It illustrates how a seemingly complex problem can be broken into simpler, more manageable steps.

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Most popular questions from this chapter

A polynomial in the variable \(x\) is defined as an expression of the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0} .\) Explain what this means.

The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. a. Write and simplify the polynomial representing \(I+P\). b. Interpret the meaning of the polynomial from part (a). c. Evaluate the polynomial \(I+P\) for \(x=6,\) and interpret the meaning of this value.

Simplify each expression. $$ \left(z^{n}+w^{m}\right)^{2} $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (\sqrt{x+1}-5)^{2} $$

If \(n>0,\) then \(-|n|=\) _____ .
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