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Chapter 4: Problem 97

Simplify. $$(x+3)(2 x-3)$$

Short Answer

Expert verified
The simplified form of the expression \((x+3)(2x-3)\) is \(2x^2 +3x -9\).

Step by step solution

01

Distribute the terms

First distribute the terms of the first binomial by the terms of the second binomial. This means to multiply the first term of the first binomial (\(x\)) by both terms of the second binomial (\(2x\) and \(-3\)), and then do the same for the second term of the first binomial (\(3\)). According to the FOIL method, you have: \((x)(2x)+(x)(-3)+ (3)(2x)+ (3)(-3)\)
02

Perform the multiplications

Now, perform the multiplication operations: \(2x^2 -3x +6x -9\)
03

Combine like terms

Finally, combine the like terms. The like terms in this expression are \(-3x\) and \(6x\). Combining these gives you: \(2x^2 +3x -9\)

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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 is a fundamental principle in mathematics that allows us to multiply a single term across terms inside parentheses. Think of it as a method to "distribute" the effect of one term over an addition or subtraction inside a bracket. Here's how it works: when you have an expression like
  • a(b + c),
you multiply 'a' by each of the terms inside the parentheses separately.
  • This turns into ab + ac.
In our example
  • you start with (x + 3)(2x - 3).
  • Apply the distributive property by taking 'x' and multiplying it by '2x' and '-3'.
  • Then take '3' from the first binomial and multiply it by '2x' and '-3'.
This step crucially expands the expression so you can simplify it later. It ensures each term gets multiplied properly, laying a foundation for the next steps of simplification.
FOIL Method
The FOIL method is a handy way to remember how to apply the distributive property when multiplying two binomials. The acronym FOIL stands for
  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in the product.
  • Inner: Multiply the inner terms.
  • Last: Multiply the last terms in each binomial.
Let's apply this to our scenario:
  • For (x+3)(2x-3), you do the First: x * 2x = 2x虏.
  • Next, Outer: x * -3 = -3x.
  • Then, Inner: 3 * 2x = 6x.
  • Last, 3 * -3 = -9.
After using the FOIL method, you end up with: 2x虏 - 3x + 6x - 9. This organized approach helps you avoid missing any terms during multiplication.
Like Terms
In algebra, like terms are terms that contain the same variables raised to the same power. They are important for simplifying expressions because they can be combined through addition or subtraction. Recognizing and combining like terms simplify the work and make expressions easier to solve. In our polynomial expression 2x虏 - 3x + 6x - 9, look for terms with the same variable and exponent.
  • The terms -3x and 6x are like terms;
  • Both have the variable 'x' raised to the first power.
You can add these together: -3x + 6x = 3x. This leaves the simplified expression as: 2x虏 + 3x - 9. Recognizing and combining like terms is a vital skill in reducing expressions to their simplest form.

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