Find the Product (2x+y-4z)^2
The question is asking for the calculation of the square of a trinomial expression. Specifically, you are being asked to find the product that results from squaring the expression (2x + y - 4z). Squaring this expression involves using algebraic methods such as the distributive property (FOIL - which stands for First, Outer, Inner, Last - in the case of binomials) or the expansion of a binomial squared to apply it to a trinomial. The resulting product is a quadratic expression in terms of x, y, and z.
$\left(\left(\right. 2 x + y - 4 z \left.\right)\right)^{2}$
Answer
Solution:
Step 1:
Express $(2x + y - 4z)^2$ as $(2x + y - 4z)(2x + y - 4z)$.
Step 2:
Distribute each term in the first binomial across the second binomial: $2x(2x) + 2x(y) + 2x(-4z) + y(2x) + y(y) + y(-4z) - 4z(2x) - 4z(y) - 4z(-4z)$.
Step 3:
Simplify each term.
Step 3.1:
Apply the commutative property to rearrange terms: $2 \cdot 2x \cdot x + 2xy + 2x(-4z) + y(2x) + y \cdot y + y(-4z) - 4z(2x) - 4zy - 4z(-4z)$.
Step 3.2:
Combine like terms by adding exponents.
Step 3.2.1:
Rearrange $x$: $2 \cdot 2(x \cdot x) + 2xy + 2x(-4z) + y(2x) + y \cdot y + y(-4z) - 4z(2x) - 4zy - 4z(-4z)$.
Step 3.2.2:
Multiply $x$ by $x$: $2 \cdot 2x^2 + 2xy + 2x(-4z) + y(2x) + y \cdot y + y(-4z) - 4z(2x) - 4zy - 4z(-4z)$.
Step 3.3:
Multiply $2$ by $2$: $4x^2 + 2xy + 2x(-4z) + y(2x) + y^2 + y(-4z) - 4z(2x) - 4zy - 4z(-4z)$.
Step 3.4:
Apply the commutative property: $4x^2 + 2xy - 2 \cdot 4xz + y(2x) + y^2 + y(-4z) - 4z(2x) - 4zy - 4z(-4z)$.
Step 3.5:
Multiply $2$ by $-4$: $4x^2 + 2xy - 8xz + y(2x) + y^2 + y(-4z) - 4z(2x) - 4zy - 4z(-4z)$.
Step 3.6:
Apply the commutative property: $4x^2 + 2xy - 8xz + 2yx + y^2 - 4yz - 4z(2x) - 4zy - 4z(-4z)$.
Step 3.7:
Multiply $y$ by $y$: $4x^2 + 2xy - 8xz + 2yx + y^2 - 4yz - 4z(2x) - 4zy - 4z(-4z)$.
Step 3.8:
Apply the commutative property: $4x^2 + 2xy - 8xz + 2yx + y^2 - 4yz - 4 \cdot 2zx - 4zy - 4z(-4z)$.
Step 3.9:
Multiply $-4$ by $2$: $4x^2 + 2xy - 8xz + 2yx + y^2 - 4yz - 8zx - 4zy - 4z(-4z)$.
Step 3.10:
Apply the commutative property: $4x^2 + 2xy - 8xz + 2yx + y^2 - 4yz - 8zx - 4zy - 4 \cdot -4z \cdot z$.
Step 3.11:
Multiply $z$ by $z$ by adding exponents.
Step 3.11.1:
Rearrange $z$: $4x^2 + 2xy - 8xz + 2yx + y^2 - 4yz - 8zx - 4zy - 4 \cdot -4(z \cdot z)$.
Step 3.11.2:
Multiply $z$ by $z$: $4x^2 + 2xy - 8xz + 2yx + y^2 - 4yz - 8zx - 4zy - 4 \cdot -4z^2$.
Step 3.12:
Multiply $-4$ by $-4$: $4x^2 + 2xy - 8xz + 2yx + y^2 - 4yz - 8zx - 4zy + 16z^2$.
Step 4:
Combine like terms $2xy$ and $2yx$.
Step 4.1:
Rearrange $y$: $4x^2 + 2xy + 2yx - 8xz + y^2 - 4yz - 8zx - 4zy + 16z^2$.
Step 4.2:
Add $2xy$ and $2yx$: $4x^2 + 4xy - 8xz + y^2 - 4yz - 8zx - 4zy + 16z^2$.
Step 5:
Combine like terms $-8zx$ and $-8xz$.
Step 5.1:
Rearrange $z$: $4x^2 + 4xy + y^2 - 4yz - 8xz - 8zx - 4zy + 16z^2$.
Step 5.2:
Combine $-8xz$ and $-8zx$: $4x^2 + 4xy + y^2 - 4yz - 16xz - 4zy + 16z^2$.
Step 6:
Combine like terms $-4zy$ and $-4yz$.
Step 6.1:
Rearrange $z$: $4x^2 + 4xy + y^2 - 4yz - 4zy - 16xz + 16z^2$.
Step 6.2:
Combine $-4yz$ and $-4zy$: $4x^2 + 4xy + y^2 - 8yz - 16xz + 16z^2$.
The final expanded form of $(2x + y - 4z)^2$ is $4x^2 + 4xy - 8yz - 16xz + y^2 + 16z^2$.
Knowledge Notes:
Binomial Expansion: The process of expanding a binomial raised to a power involves using the distributive property to multiply each term in the first binomial by each term in the second binomial.
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not affect the product, i.e., $ab = ba$.
Multiplying Exponents: When multiplying like bases, the exponents are added, i.e., $x^a \cdot x^b = x^{a+b}$.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power.
Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses, i.e., $a(b + c) = ab + ac$.
LaTeX Formatting: In the solution, LaTeX is used to format mathematical expressions, ensuring that they are clearly presented and easy to read. For example, $x^2$ is written in LaTeX as $x^2$.
