Transform f(x) = -x(x-8) to Its Standard Representation

Q: Why does (-x) times (-8) equal +8x?

Remember the rule: negative times negative equals positive. When you have (-x) × (-8), you're multiplying two negative values, so the result is positive: +8x.

Q: What's the difference between factored and standard form?

Factored form shows factors like -x(x-8), while standard form shows the polynomial expanded: $ -x^2 + 8x $. Both represent the same function!

Q: Do I always need to write terms in order of decreasing powers?

Yes! In standard form, arrange terms from highest degree to lowest. So $ -x^2 + 8x $ is correct, not $ 8x - x^2 $.

Q: How can I check if my expansion is correct?

Pick any value for x and substitute it into both the original and expanded forms. If they give the same result, your expansion is correct!

Polynomial Expansion with Distributive Property

Find the standard representation of the following function

$f(x)=-x(x-8)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplified to the standard representation of the function

00:03 Open parentheses properly, multiply by each factor

00:08 Calculate multiplications

00:11 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

$f(x)=-x(x-8)$

Step-by-step solution

To solve this problem, we'll convert the given function from its factored form to the standard form using the distributive property. The given function is $f(x) = -x(x - 8)$ .

Let's go through the necessary steps:

Step 1: Apply the distributive property to expand the expression.
$f(x) = -x(x - 8) = -x \cdot x + (-x) \cdot (-8)$
Step 2: Simplify each term.
$-x \cdot x = -x^2$ and $(-x) \cdot (-8) = +8x$ .
Step 3: Combine the terms to express $f(x)$ in standard form:
$f(x) = -x^2 + 8x$ .

Therefore, the standard representation of the function is $f(x) = -x^2 + 8x$ .

Comparing this result to the multiple-choice options, we can see that the correct choice is option 3: $f(x)=-x^2+8x$ .

Final Answer

$f(x)=-x^2+8x$

Key Points to Remember

Essential concepts to master this topic

Distributive Property: Multiply each term inside parentheses by the outside term
Technique: $-x(x-8) = -x \cdot x + (-x) \cdot (-8) = -x^2 + 8x$
Check: Substitute x = 2: $-2(2-8) = -2(-6) = 12$ and $-4 + 16 = 12$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative sign when distributing
Don't distribute as $-x(x-8) = -x^2 - 8x$ ! This ignores that (-x) × (-8) = +8x, not -8x. Always track negative signs carefully: the negative outside times the negative 8 gives positive 8x.

Practice Quiz

Test your knowledge with interactive questions

Create an algebraic expression based on the following parameters:

$$ a=3,b=0,c=-3 $$

FAQ

Everything you need to know about this question

Why does (-x) times (-8) equal +8x?

Remember the rule: negative times negative equals positive. When you have (-x) × (-8), you're multiplying two negative values, so the result is positive: +8x.

What's the difference between factored and standard form?

Factored form shows factors like -x(x-8), while standard form shows the polynomial expanded: $-x^2 + 8x$ . Both represent the same function!

Do I always need to write terms in order of decreasing powers?

Yes! In standard form, arrange terms from highest degree to lowest. So $-x^2 + 8x$ is correct, not $8x - x^2$ .

How can I check if my expansion is correct?

Pick any value for x and substitute it into both the original and expanded forms. If they give the same result, your expansion is correct!

What if I have more than two terms in the parentheses?

Use the same distributive property! Multiply the outside term by each term inside the parentheses, then combine like terms.

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Transform f(x) = -x(x-8) to Its Standard Representation

Polynomial Expansion with Distributive Property

❤️ 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 does (-x) times (-8) equal +8x?

What's the difference between factored and standard form?

Do I always need to write terms in order of decreasing powers?

How can I check if my expansion is correct?

What if I have more than two terms in the parentheses?

🌟 Unlock Your Math Potential

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