Expand the Expression: Finding (x-x²)² Step by Step

Q: Why can't I just square each term separately?

Because (a - b)² ≠ a² - b²! You're missing the cross-multiplication term. The binomial formula ensures you include all the products when expanding.

Q: How do I remember which term gets the coefficient -2?

The middle term -2ab always comes from multiplying the first and second terms together twice. In (x - x²)², this gives -2(x)(x²) = -2x³.

Q: What if the signs were different, like (x + x²)²?

Use (a + b)² = a² + 2ab + b² instead. The middle term would be positive: +2x³, giving you x⁴ + 2x³ + x².

Q: Can I check my answer by multiplying (x - x²)(x - x²)?

Absolutely! Use FOIL or distribution: First terms (x·x = x²), Outer terms (x·(-x²) = -x³), Inner terms ((-x²)·x = -x³), Last terms ((-x²)·(-x²) = x⁴). Sum: x⁴ - 2x³ + x².

Q: Why is the highest power x⁴ instead of x²?

When you square a polynomial, the degree doubles. Since x² is degree 2, squaring the binomial gives you degree 4. The term (x²)² = x⁴ creates the highest power.

Binomial Expansion with Polynomial Terms

$(x-x^2)^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's make it simple.

00:08 We'll use shortened multiplication formulas to expand the parentheses.

00:17 Here, the letter X represents A in our formula.

00:24 And X squared represents B in the formula.

00:31 So, let's substitute into the formula and solve together.

00:46 Time to solve the multiplications step by step.

00:53 Great job! And that's how we find the solution to the problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(x-x^2)^2=$

Step-by-step solution

To solve the expression $(x-x^2)^2$ , we will use the square of a binomial formula $(a-b)^2 = a^2 - 2ab + b^2$ .

Let's identify $a$ and $b$ in our expression:

Here, $a = x$ and $b = x^2$ .

Applying the formula:

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

Calculating each part, we get:

$(x)^2 = x^2$
$-2(x)(x^2) = -2x^3$
$(x^2)^2 = x^4$

Combining these results, the expression simplifies to:

$x^4 - 2x^3 + x^2$

Therefore, the expanded form of $(x-x^2)^2$ is $\boxed{x^4 - 2x^3 + x^2}$ .

Final Answer

$x^4-2x^3+x^2$

Key Points to Remember

Essential concepts to master this topic

Formula: Use (a-b)² = a² - 2ab + b² for binomial squares
Technique: With a = x and b = x², calculate (x)² - 2(x)(x²) + (x²)²
Check: Verify x⁴ - 2x³ + x² factors back to (x - x²)² ✓

Common Mistakes

Avoid these frequent errors

Forgetting the middle term coefficient
Don't expand (x - x²)² as x² - x⁴ = missing the -2ab term! This ignores the cross-multiplication and gives incomplete results. Always apply the full formula a² - 2ab + b² with the coefficient -2.

Practice Quiz

Test your knowledge with interactive questions

Declares the given expression as a sum

$$ (7b-3x)^2 $$

FAQ

Everything you need to know about this question

Why can't I just square each term separately?

Because (a - b)² ≠ a² - b²! You're missing the cross-multiplication term. The binomial formula ensures you include all the products when expanding.

How do I remember which term gets the coefficient -2?

The middle term -2ab always comes from multiplying the first and second terms together twice. In (x - x²)², this gives -2(x)(x²) = -2x³.

What if the signs were different, like (x + x²)²?

Use (a + b)² = a² + 2ab + b² instead. The middle term would be positive: +2x³, giving you x⁴ + 2x³ + x².

Can I check my answer by multiplying (x - x²)(x - x²)?

Absolutely! Use FOIL or distribution: First terms (x·x = x²), Outer terms (x·(-x²) = -x³), Inner terms ((-x²)·x = -x³), Last terms ((-x²)·(-x²) = x⁴). Sum: x⁴ - 2x³ + x².

Why is the highest power x⁴ instead of x²?

When you square a polynomial, the degree doubles. Since x² is degree 2, squaring the binomial gives you degree 4. The term (x²)² = x⁴ creates the highest power.

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