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

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

Because (a + b)² ≠ a² + b²! When you expand a square, you get three terms: the squares of each term PLUS the cross-product term $ 2ab $. Missing this middle term is the most common mistake.

Q: How do I handle the exponents when multiplying?

Use the exponent rule: when multiplying powers with the same base, add the exponents. So $ x \cdot x^2 = x^{1+2} = x^3 $ and $ (x^2)^2 = x^{2 \times 2} = x^4 $.

Q: What order should I write my final answer?

Write terms in ascending order of degree (lowest power first): $ x^2 + 2x^3 + x^4 $. This makes it easier to read and matches standard polynomial form.

Q: Can I use FOIL instead of the binomial formula?

Yes! FOIL works perfectly here. First: x·x = x², Outer: x·x² = x³, Inner: x²·x = x³, Last: x²·x² = x⁴. Then combine: x² + x³ + x³ + x⁴ = x² + 2x³ + x⁴.

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

Pick a simple value like x = 1. Original: $ (1 + 1^2)^2 = 2^2 = 4 $. Expanded: $ 1^2 + 2(1)^3 + 1^4 = 1 + 2 + 1 = 4 $ ✓. They match!

Binomial Expansion with Variable Exponents

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

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Step-by-step video solution

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00:00 Simply

00:10 We will use the shortened multiplication formulas

00:41 And this is the solution to the question

Step-by-step written solution

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Understand the problem

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

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Identify the terms: Here, $a = x$ and $b = x^2$ .
Step 2: Apply the square of a binomial formula: $(a + b)^2 = a^2 + 2ab + b^2$ .
Step 3: Substitute the terms into the formula and simplify the resulting expression.

Now, let's work through each step:

Step 1: We have $a = x$ and $b = x^2$ .
Step 2: The formula gives us: $(a + b)^2 = a^2 + 2ab + b^2$

Step 3: Substituting $a = x$ and $b = x^2$ into the formula:

(x + x^2)^2 = (x)^2 + 2(x)(x^2) + (x^2)^2

This simplifies to:

x^2 + 2x^3 + x^4

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

This matches with choice 3 from the options provided.

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Formula: (a + b)² = a² + 2ab + b² for any terms a and b
Technique: Substitute a = x and b = x² to get x² + 2(x)(x²) + (x²)²
Check: Final answer x² + 2x³ + x⁴ has terms in ascending degree order ✓

Common Mistakes

Avoid these frequent errors

Squaring each term separately without the middle term
Don't calculate (x + x²)² = x² + (x²)² = x² + x⁴! This ignores the crucial middle term 2ab and gives a completely wrong result. Always use the complete binomial formula (a + b)² = a² + 2ab + b² to include all three terms.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+3)^2 $$

FAQ

Everything you need to know about this question

Why can't I just square each term separately?

Because (a + b)² ≠ a² + b²! When you expand a square, you get three terms: the squares of each term PLUS the cross-product term $2ab$ . Missing this middle term is the most common mistake.

How do I handle the exponents when multiplying?

Use the exponent rule: when multiplying powers with the same base, add the exponents. So $x \cdot x^2 = x^{1+2} = x^3$ and $(x^2)^2 = x^{2 \times 2} = x^4$ .

What order should I write my final answer?

Write terms in ascending order of degree (lowest power first): $x^2 + 2x^3 + x^4$ . This makes it easier to read and matches standard polynomial form.

Can I use FOIL instead of the binomial formula?

Yes! FOIL works perfectly here. First: x·x = x², Outer: x·x² = x³, Inner: x²·x = x³, Last: x²·x² = x⁴. Then combine: x² + x³ + x³ + x⁴ = x² + 2x³ + x⁴.

How can I check if my expansion is correct?

Pick a simple value like x = 1. Original: $(1 + 1^2)^2 = 2^2 = 4$ . Expanded: $1^2 + 2(1)^3 + 1^4 = 1 + 2 + 1 = 4$ ✓. They match!

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