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Chapter 0: Problem 97

Use the Distributive Property to remove the parentheses. $$ (x+9)(2 x-7) $$

Short Answer

Expert verified
2x^2 + 11x - 63

Step by step solution

01

Apply the Distributive Property

The Distributive Property states that for any numbers or expressions a, b, and c, the expression a(b + c) = ab + ac. Use this property to distribute each term in the first parenthesis over each term in the second parenthesis.
02

Distribute the first term

Multiply the first term of the first parenthesis (x) by each term in the second parenthesis: (x) *(2x - 7). This results in: (x)(2x) + (x)(-7) = 2x^2 - 7x.
03

Distribute the second term

Now, multiply the second term of the first parenthesis (9) by each term in the second parenthesis: (9) *(2x - 7). This results in: (9)(2x) + (9)(-7) = 18x - 63.
04

Combine all terms

Add the results from Step 2 and Step 3 together: 2x^2 - 7x + 18x - 63. Combine like terms to simplify: 2x^2 + 11x - 63.

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Key Concepts

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

Polynomial Multiplication
When you are multiplying polynomials, each term in the first polynomial needs to be multiplied by every term in the second polynomial. This process is known as using the Distributive Property. For example, consider the expression \((x+9)(2x-7)\). To multiply these polynomials, follow these steps:

  • First, distribute the first term in the first polynomial \(x\) to each term in the second polynomial \(2x-7\): \(x(2x) + x(-7) = 2x^2 - 7x\).

  • Then, distribute the second term in the first polynomial \(9\) to each term in the second polynomial \(2x-7\): \(9(2x) + 9(-7) = 18x - 63\).

  • Add the results from both distributions together: \(2x^2 - 7x + 18x - 63\).

This method ensures that every term is accounted for and properly multiplied.
Combining Like Terms
After you have multiplied the polynomials, you need to combine like terms to simplify the expression. Like terms are terms that have the same variables raised to the same power. For example, in the polynomial expression \(2x^2 - 7x + 18x - 63\), the like terms are \(-7x\) and \(18x\).

To combine them, you simply add or subtract their coefficients. In this case, you would add \(-7x\) and \(18x\):

\[-7x + 18x = 11x\]

Therefore, when you combine like terms, the expression \((x+9)(2x-7)\) simplifies to \(2x^2 + 11x - 63\).
Simplifying Expressions
The final step in the process is to simplify the expression. Once you have combined all the like terms, you might have an expression where no further simplification is needed. For instance, in our example, after combining \(-7x\) and \(18x\), we got \((2x^2 + 11x - 63)\).

Simplifying an expression means:
  • Removing any parentheses by using the Distributive Property.

  • Combining all like terms.

  • Ensuring that no further operations can be performed.

In our example, we accomplished all steps and reached the simplest form of the expression: \(2x^2 + 11x - 63\). Always double-check that every term has been handled and that no more combinations or operations can be performed.

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