Convert the Function f(x) = (2x + 1)(x - 2) into Standard Form

Polynomial Expansion with FOIL Method

Find the standard representation of the following function

f(x)=(2x+1)(x2) f(x)=(2x+1)(x-2)

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Expanded to the standard representation of the function
00:03 Open parentheses properly, multiply each factor by each factor
00:14 Collect terms
00:24 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Find the standard representation of the following function

f(x)=(2x+1)(x2) f(x)=(2x+1)(x-2)

2

Step-by-step solution

To find the standard representation of the function f(x)=(2x+1)(x2) f(x) = (2x + 1)(x - 2) , we'll follow these steps to expand and simplify the expression:

  • Step 1: Distribute each term of the first binomial over each term of the second binomial using the FOIL method.
  • Step 2: Combine like terms to express the function in standard quadratic form.

Now, let's expand the expression:
1. Multiply the first terms: 2xx=2x2 2x \cdot x = 2x^2
2. Multiply the outer terms: 2x(2)=4x 2x \cdot (-2) = -4x
3. Multiply the inner terms: 1x=x 1 \cdot x = x
4. Multiply the last terms: 1(2)=2 1 \cdot (-2) = -2

Next, we combine these results:
- The 2x2 2x^2 term remains as is.
- Add the linear terms: 4x+x=3x -4x + x = -3x
- The constant term is 2-2.

Thus, the expanded and simplified form of the function is:
f(x)=2x23x2 f(x) = 2x^2 - 3x - 2

The final expression in standard form is f(x)=2x23x2 f(x) = 2x^2 - 3x - 2 .

3

Final Answer

f(x)=2x23x2 f(x)=2x^2-3x-2

Key Points to Remember

Essential concepts to master this topic
  • Standard Form: Arrange terms as ax2+bx+c ax^2 + bx + c with decreasing powers
  • FOIL Method: First (2x)(x) = 2x², Outer (2x)(-2) = -4x, Inner (1)(x) = x, Last (1)(-2) = -2
  • Check: Verify by factoring 2x23x2 2x^2 - 3x - 2 returns (2x+1)(x2) (2x+1)(x-2)

Common Mistakes

Avoid these frequent errors
  • Incorrectly combining like terms after FOIL
    Don't combine -4x + x = -5x! This gives the wrong middle term coefficient. The error happens when you add instead of subtract or make sign errors. Always carefully track negative signs: -4x + x = -3x.

Practice Quiz

Test your knowledge with interactive questions

Create an algebraic expression based on the following parameters:

\( a=2,b=2,c=2 \)

FAQ

Everything you need to know about this question

What does 'standard form' mean for a quadratic function?

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Standard form means writing the function as f(x)=ax2+bx+c f(x) = ax^2 + bx + c where terms are arranged by decreasing powers of x. This makes it easy to identify the coefficients!

Why do we use the FOIL method here?

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FOIL helps you systematically multiply two binomials without missing any terms. It stands for First, Outer, Inner, Last - the four products you need to find and then add together.

How do I know which terms are 'like terms'?

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Like terms have the same variable and same power. In this problem, -4x and +x are like terms because both have x¹, so they combine to give -3x.

What if I get a different answer than the choices given?

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Double-check your FOIL multiplication and combining like terms! The most common errors are sign mistakes or forgetting to distribute negative signs properly.

Can I expand this a different way besides FOIL?

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Yes! You can use the distributive property twice: distribute (2x+1) to both x and -2, then combine. FOIL is just a organized way to do the same thing.

How can I check if my final answer is correct?

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Try factoring your answer back! If 2x23x2 2x^2 - 3x - 2 factors to (2x+1)(x2) (2x+1)(x-2) , you know it's right. You can also substitute a test value like x = 1.

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