Problem 251 Find each product. $$ \left(... [FREE SOLUTION]

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Chapter 5: Problem 251

Find each product. $$ \left(y^{4}+2 z\right)^{2} $$

Short Answer

Expert verified

y^8 + 4y^4z + 4z^2

Step by step solution

Identify the expression

The given expression is $\left(y^{4}+2 z\right)^{2}$. This is an example of a binomial squared.

Apply the binomial square formula

Use the binomial square formula for $(a + b)^2$, which is $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = y^4$ and $b = 2z$.

Square the first term

Calculate $a^2$: $a = y^4$, so $a^2 = (y^4)^2 = y^8$.

Multiply the two terms and double the product

Calculate $(2ab)$: $2 \times y^4 \times 2z = 4y^4z$.

Square the second term

Calculate $b^2$: $b = 2z$, so $b^2 = (2z)^2 = 4z^2$.

Combine the results

Combine all the terms: \[y^8 + 4y^4z + 4z^2\] is the expanded form of $(y^4 + 2z)^2$.

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Key Concepts

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

binomial square formula

In algebra, the binomial square formula is a shortcut to expand polynomials where a binomial is squared. Instead of using polynomial multiplication directly, we use the identity for squaring a binomial. This makes the process much quicker and easier. For a binomial $(a + b)^2$, the formula is: $(a + b)^2 = a^2 + 2ab + b^2$.
In our exercise, $(y^4 + 2z)^2$, we see that:

$a = y^4$
$b = 2z$

This means, when squared:

$a^2 = (y^4)^2 = y^8$
$2ab = 2 \cdot y^4 \cdot 2z = 4y^4z$
$b^2 = (2z)^2 = 4z^2$

Combining these, we get the expanded form: $y^8 + 4y^4z + 4z^2$.

polynomial multiplication

Polynomial multiplication involves distributing every term in the first polynomial to every term in the second polynomial. In our exercise, multiplying the binomial $(y^4 + 2z)$ by itself can be broken down into multiple steps. Using the distributive property:

Multiply $y^4$ by each term in $y^4 + 2z$, resulting in $(y^4)\times(y^4) + (y^4)\times(2z)$.
Multiply $2z$ by each term in $y^4 + 2z$, resulting in $(2z)\times(y^4) + (2z)\times(2z)$.

So, we get:

First part: $y^4 \times y^4 = y^8$
Second part: $y^4 \times 2z = 2y^4z$
Third part: $2z \times y^4 = 2y^4z$
Fourth part: $2z \times 2z = 4z^2$

Combining all parts, notice that $2y^4z$ and another $2y^4z$ combine to make $4y^4z$. Therefore, the expanded polynomial $y^8 + 4y^4z + 4z^2$.

algebraic expressions

Algebraic expressions consist of terms that include variables, constants, and arithmetic operations. In our exercise, the expression $(y^4 + 2z)^2$ involves:

Variables: $y$ and $z$
Constants: $2$

To handle this expression correctly, you need to understand operations like exponentiation and multiplication.
For instance:

Squaring $y^4$ results in $y^8$ because multiplying exponents: $(y^4)^2 = y^{(4 \times 2)}$.
Multiplying $y^4$ by $2z$ yields $2y^4z$, where coefficients and variables are handled separately.

Breaking down complex expressions into simpler, more manageable parts is the key skill in algebra. By mastering these, you will find working with algebraic expressions easier. The final expression $y^8 + 4y^4z + 4z^2$ in our exercise emerges clearly from following these steps.

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91影视

Find each product. $$ \left(y^{4}+2 z\right)^{2} $$

Short Answer

Step by step solution

Identify the expression

Apply the binomial square formula

Square the first term

Multiply the two terms and double the product

Square the second term

Combine the results

Key Concepts

binomial square formula

polynomial multiplication

algebraic expressions

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