Expand and Simplify: Transforming f(x) = 3x(x+4) to Standard Form

Q: Why can't I just add 3 to each term in the parentheses?

Because multiplication and addition are completely different operations! The expression $ 3x(x+4) $ means 3x is multiplied by the entire quantity (x+4), not added to it.

Q: What's the difference between 3x(x+4) and 3(x²+4x)?

They're actually the same thing! In the first form, you need to expand x(x+4) first to get x²+4x, then multiply by 3. Both give you $ 3x^2 + 12x $.

Q: How do I know which term gets multiplied by what?

Use the distributive property: the term outside the parentheses (3x) multiplies every single term inside the parentheses. So 3x × x = 3x² and 3x × 4 = 12x.

Q: Why is there no constant term in the final answer?

Because the original expression $ 3x(x+4) $ doesn't have any standalone numbers - only terms with x. When you expand, you get 3x² + 12x + 0, so the constant term is zero.

Q: Can I check my answer by plugging in a number for x?

Absolutely! Try x = 2: Original gives 3(2)(2+4) = 6(6) = 36. Your answer gives 3(4) + 12(2) = 12 + 24 = 36. If they match, you're correct!

Distributive Property with Quadratic Expansion

Find the standard representation of the following function

$f(x)=3x(x+4)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify for the standard representation of the function

00:03 Open parentheses properly, multiply by each factor

00:15 Calculate multiplications

00:20 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the standard representation of the following function

$f(x)=3x(x+4)$

Step-by-step solution

To find the standard representation of the quadratic function $f(x) = 3x(x + 4)$ , follow these steps:

Step 1: Apply the distributive property to expand the expression $x(x + 4)$ .
Using this property, we have:
$x(x + 4) = x \cdot x + x \cdot 4 = x^2 + 4x$ .
Step 2: Multiply each term by the coefficient outside the parenthesis, which is 3.
This gives us:
$3(x^2 + 4x) = 3 \cdot x^2 + 3 \cdot 4x$ .
Step 3: Simplify by performing the multiplication.
$3x^2 + 12x$ .

Therefore, the standard representation of the function is $f(x) = 3x^2 + 12x$ . This matches choice 3 in the provided answers.

Final Answer

$f(x)=3x^2+12x$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply distributive property to multiply each term inside parentheses
Technique: 3x(x+4) becomes 3x·x + 3x·4 = 3x² + 12x
Check: Factor result back: 3x² + 12x = 3x(x + 4) ✓

Common Mistakes

Avoid these frequent errors

Adding instead of multiplying the outer coefficient
Don't add the 3 to get 3x² + 4x + 3 = wrong coefficients! This ignores the distributive property completely. Always multiply the outer coefficient by every term inside the parentheses.

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

Why can't I just add 3 to each term in the parentheses?

Because multiplication and addition are completely different operations! The expression $3x(x+4)$ means 3x is multiplied by the entire quantity (x+4), not added to it.

What's the difference between 3x(x+4) and 3(x²+4x)?

They're actually the same thing! In the first form, you need to expand x(x+4) first to get x²+4x, then multiply by 3. Both give you $3x^2 + 12x$ .

How do I know which term gets multiplied by what?

Use the distributive property: the term outside the parentheses (3x) multiplies every single term inside the parentheses. So 3x × x = 3x² and 3x × 4 = 12x.

Why is there no constant term in the final answer?

Because the original expression $3x(x+4)$ doesn't have any standalone numbers - only terms with x. When you expand, you get 3x² + 12x + 0, so the constant term is zero.

Can I check my answer by plugging in a number for x?

Absolutely! Try x = 2: Original gives 3(2)(2+4) = 6(6) = 36. Your answer gives 3(4) + 12(2) = 12 + 24 = 36. If they match, you're correct!

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Expand and Simplify: Transforming f(x) = 3x(x+4) to Standard Form

Distributive Property with Quadratic Expansion

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just add 3 to each term in the parentheses?

What's the difference between 3x(x+4) and 3(x²+4x)?

How do I know which term gets multiplied by what?

Why is there no constant term in the final answer?

Can I check my answer by plugging in a number for x?

🌟 Unlock Your Math Potential

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