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Chapter 10: Problem 41

Multiply the polynomials. $$(x+y)^{2}$$

Short Answer

Expert verified
\((x+y)^{2} = x^{2} + 2xy + y^{2} \).

Step by step solution

01

- Write the polynomial in expanded form

To start, recognize that \((x+y)^{2}\) is a polynomial squared. This can be written as \((x+y) \times (x+y)\).
02

- Distribute each term

Next, distribute each term in the first polynomial to every term in the second polynomial as follows:\[ (x+y) \times (x+y) = x \times x + x \times y + y \times x + y \times y \]
03

- Simplify the multiplication

Then, perform the multiplication for each term: \[ x \times x + x \times y + y \times x + y \times y = x^{2} + xy + yx + y^{2} \]
04

- Combine like terms

Finally, combine the like terms \((xy \text{ and } yx)\):\[ x^{2} + xy + yx + y^{2} = x^{2} + 2xy + y^{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Expansion
When we talk about polynomial expansion, it means expressing a polynomial that needs to be multiplied out in its extended form. For example, \((x+y)^{2}\) can be expanded to \((x+y) \times (x+y)\). This step reveals all the products that need to be computed. Expanding helps to see each term that will be multiplied. It鈥檚 like unwrapping a package to see what鈥檚 inside. This makes it easier to manage the multiplication process. Once expanded, the polynomial can be simplified step by step.
Distributive Property
The distributive property is a crucial rule in algebra. It allows us to break down expressions like \((x+y) \times (x+y)\). The idea is to distribute each term in one polynomial to every term in the other polynomial. Here鈥檚 how it works:
  • First, take the \(x\) from \(x+y\) and multiply it by both terms in the second \(x+y\).
  • Then, take the \(y\) in \(x+y\) and multiply it by both terms in the other \(x+y\).
\((x+y) \times (x+y)\) becomes \(x \times x + x \times y + y \times x + y \times y\). Distributing in this way ensures that each possible product is considered.
Combining Like Terms
Combining like terms means adding or subtracting terms that have the same variable raised to the same power. In our example \(x^{2} + xy + yx + y^{2}\):
  • \(xy\) and \(yx\) are like terms because they both have the same variable parts.
We combine these to get \(2xy\). Therefore, \(x^{2} + xy + yx + y^{2}\) simplifies to \(x^{2} + 2xy + y^{2}\). This step makes your expression more compact and easier to understand.
Binomial Theorem
The binomial theorem provides a formula for expanding any power of a binomial without expanding step-by-step. It states that: \[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\] For \((x+y)^{2}\), it looks like this: \[ (x+y)^{2} = \binom{2}{0}x^{2}y^{0} + \binom{2}{1}x^{1}y^{1} + \binom{2}{2}x^{0}y^{2}
\] Calculating the binomial coefficients \((\binom{n}{k})\), we get: \[ x^{2} + 2xy + y^{2} \] This matches our expanded and simplified form. The binomial theorem is especially helpful for expanding higher powers of polynomials efficiently.

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Most popular questions from this chapter

Perform the indicated operations. $$\left(-5.1 y^{2}+4.8 y+2.3\right)+\left(-1.1 y^{2}-8.9 y+3.0\right)$$

Simplify. Write the answers with positive exponents only. $$5^{4} 5^{-7}$$

Perform the indicated operations. $$\left(-5 a^{2}-9 a^{3}+10 a^{4}\right)+\left(2 a^{3}-a^{4}\right)-\left(8 a+4 a^{3}+10 a^{4}\right)$$

Simplify. Write the answer with positive exponents only. $$\frac{5}{a^{-2} b^{-4}}$$

Simplify. Write the answers with positive exponents only. $$\frac{2^{3}}{2^{-2}}$$
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