Complete the Polynomial Equation: Solve for x and y Blanks in (x^2+y)^2+(y^2+x)^2

Q: Why do I need to expand both squared terms separately?

Each squared binomial has its own expansion pattern. Expanding separately ensures you capture all terms correctly before combining like terms.

Q: How do I identify what goes in each blank?

Compare your expanded result with the given form. Match coefficients and variable patterns to determine what belongs in each blank position.

Q: What if my expanded terms don't seem to match the pattern?

Double-check your expansion using $ (a+b)^2 = a^2 + 2ab + b^2 $. Then group like terms carefully - you might need to factor out common expressions.

Q: Why is the middle term 2xy(x+y)?

This comes from combining the cross terms: $ 2x^2y + 2y^2x = 2xy(x+y) $. Factoring out common factors helps identify the pattern.

Polynomial Expansion with Term Identification

Fill in the blanks:

$(x^2+y)^2+(y^2+x)^2=?(x^2+1)+2xy\lbrack?\rbrack+y^2\lbrack?\rbrack$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing

00:04 We will use the shortened multiplication formulas to open the parentheses

00:19 We will calculate the squares and products

00:37 We will find common factors and factor out of parentheses

01:24 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

Fill in the blanks:

$(x^2+y)^2+(y^2+x)^2=?(x^2+1)+2xy\lbrack?\rbrack+y^2\lbrack?\rbrack$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Expand the squares $(x^2+y)^2$ and $(y^2+x)^2$ .
Step 2: Combine like terms from the expansions.
Step 3: Compare the result to the right-hand side of the equation.

Now, let's work through each step:

Step 1: Expand the expressions:
$(x^2 + y)^2 = (x^2)^2 + 2(x^2)(y) + y^2 = x^4 + 2x^2y + y^2$
$(y^2 + x)^2 = (y^2)^2 + 2(y^2)(x) + x^2 = y^4 + 2y^2x + x^2$

Step 2: Combine the expansions:
$(x^2+y)^2 + (y^2+x)^2 = x^4 + 2x^2y + y^2 + y^4 + 2y^2x + x^2$
Combine like terms:
$= x^4 + x^2 + y^4 + y^2 + 2xy(x+y)$

Step 3: Compare with $(x^2+1) + 2xy\lbrack?\rbrack + y^2\lbrack?\rbrack$ :
The complete expression terms include $(x^2+1)$ , a term linked to $2xy(x+y)$ , and a term in $y^2$ which should account for $y^4 + y^2$ similarity.

Therefore, the solution to the problem is that the blanks should be filled with $x^2,(x+y),y^2+1$ .

Final Answer

$x^2,(x+y),y^2+1$

Key Points to Remember

Essential concepts to master this topic

Expansion Rule: Use $(a+b)^2 = a^2 + 2ab + b^2$ for each squared term
Technique: Group like terms: $x^4 + x^2 + y^4 + y^2 + 2xy(x+y)$
Check: Verify coefficient patterns match the given form structure ✓

Common Mistakes

Avoid these frequent errors

Incorrectly expanding squared binomials
Don't forget the middle term when expanding $(x^2+y)^2$ = wrong coefficients! Missing $2x^2y$ leads to incomplete expansion. Always remember $(a+b)^2 = a^2 + 2ab + b^2$ includes the cross term.

Practice Quiz

Test your knowledge with interactive questions

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

$$ (x+y)^2 $$

FAQ

Everything you need to know about this question

Why do I need to expand both squared terms separately?

Each squared binomial has its own expansion pattern. Expanding separately ensures you capture all terms correctly before combining like terms.

How do I identify what goes in each blank?

Compare your expanded result with the given form. Match coefficients and variable patterns to determine what belongs in each blank position.

What if my expanded terms don't seem to match the pattern?

Double-check your expansion using $(a+b)^2 = a^2 + 2ab + b^2$ . Then group like terms carefully - you might need to factor out common expressions.

How can I verify my answer is correct?

Substitute your values back into the blanks and expand the right side. It should equal your original expanded form: $x^4 + x^2 + y^4 + y^2 + 2xy(x+y)$ .

Why is the middle term 2xy(x+y)?

This comes from combining the cross terms: $2x^2y + 2y^2x = 2xy(x+y)$ . Factoring out common factors helps identify the pattern.

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