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.

${((2 x + y - 4 z))}^{2}$

Answer

Expert–verified

Solution:

Step 1:

Express $(2 x + y - 4 z)^{2}$ as $(2 x + y - 4 z) (2 x + y - 4 z)$ .

Step 2:

Distribute each term in the first binomial across the second binomial: $2 x (2 x) + 2 x (y) + 2 x (- 4 z) + y (2 x) + y (y) + y (- 4 z) - 4 z (2 x) - 4 z (y) - 4 z (- 4 z)$ .

Step 3:

Simplify each term.

Step 3.1:

Apply the commutative property to rearrange terms: $2 \cdot 2 x \cdot x + 2 x y + 2 x (- 4 z) + y (2 x) + y \cdot y + y (- 4 z) - 4 z (2 x) - 4 z y - 4 z (- 4 z)$ .

Step 3.2:

Combine like terms by adding exponents.

Step 3.2.1:

Rearrange $x$ : $2 \cdot 2 (x \cdot x) + 2 x y + 2 x (- 4 z) + y (2 x) + y \cdot y + y (- 4 z) - 4 z (2 x) - 4 z y - 4 z (- 4 z)$ .

Step 3.2.2:

Multiply $x$ by $x$ : $2 \cdot 2 x^{2} + 2 x y + 2 x (- 4 z) + y (2 x) + y \cdot y + y (- 4 z) - 4 z (2 x) - 4 z y - 4 z (- 4 z)$ .

Step 3.3:

Multiply $2$ by $2$ : $4 x^{2} + 2 x y + 2 x (- 4 z) + y (2 x) + y^{2} + y (- 4 z) - 4 z (2 x) - 4 z y - 4 z (- 4 z)$ .

Step 3.4:

Apply the commutative property: $4 x^{2} + 2 x y - 2 \cdot 4 x z + y (2 x) + y^{2} + y (- 4 z) - 4 z (2 x) - 4 z y - 4 z (- 4 z)$ .

Step 3.5:

Multiply $2$ by $- 4$ : $4 x^{2} + 2 x y - 8 x z + y (2 x) + y^{2} + y (- 4 z) - 4 z (2 x) - 4 z y - 4 z (- 4 z)$ .

Step 3.6:

Apply the commutative property: $4 x^{2} + 2 x y - 8 x z + 2 y x + y^{2} - 4 y z - 4 z (2 x) - 4 z y - 4 z (- 4 z)$ .

Step 3.7:

Multiply $y$ by $y$ : $4 x^{2} + 2 x y - 8 x z + 2 y x + y^{2} - 4 y z - 4 z (2 x) - 4 z y - 4 z (- 4 z)$ .

Step 3.8:

Apply the commutative property: $4 x^{2} + 2 x y - 8 x z + 2 y x + y^{2} - 4 y z - 4 \cdot 2 z x - 4 z y - 4 z (- 4 z)$ .

Step 3.9:

Multiply $- 4$ by $2$ : $4 x^{2} + 2 x y - 8 x z + 2 y x + y^{2} - 4 y z - 8 z x - 4 z y - 4 z (- 4 z)$ .

Step 3.10:

Apply the commutative property: $4 x^{2} + 2 x y - 8 x z + 2 y x + y^{2} - 4 y z - 8 z x - 4 z y - 4 \cdot - 4 z \cdot z$ .

Step 3.11:

Multiply $z$ by $z$ by adding exponents.

Step 3.11.1:

Rearrange $z$ : $4 x^{2} + 2 x y - 8 x z + 2 y x + y^{2} - 4 y z - 8 z x - 4 z y - 4 \cdot - 4 (z \cdot z)$ .

Step 3.11.2:

Multiply $z$ by $z$ : $4 x^{2} + 2 x y - 8 x z + 2 y x + y^{2} - 4 y z - 8 z x - 4 z y - 4 \cdot - 4 z^{2}$ .

Step 3.12:

Multiply $- 4$ by $- 4$ : $4 x^{2} + 2 x y - 8 x z + 2 y x + y^{2} - 4 y z - 8 z x - 4 z y + 16 z^{2}$ .

Step 4:

Combine like terms $2 x y$ and $2 y x$ .

Step 4.1:

Rearrange $y$ : $4 x^{2} + 2 x y + 2 y x - 8 x z + y^{2} - 4 y z - 8 z x - 4 z y + 16 z^{2}$ .

Step 4.2:

Add $2 x y$ and $2 y x$ : $4 x^{2} + 4 x y - 8 x z + y^{2} - 4 y z - 8 z x - 4 z y + 16 z^{2}$ .

Step 5:

Combine like terms $- 8 z x$ and $- 8 x z$ .

Step 5.1:

Rearrange $z$ : $4 x^{2} + 4 x y + y^{2} - 4 y z - 8 x z - 8 z x - 4 z y + 16 z^{2}$ .

Step 5.2:

Combine $- 8 x z$ and $- 8 z x$ : $4 x^{2} + 4 x y + y^{2} - 4 y z - 16 x z - 4 z y + 16 z^{2}$ .

Step 6:

Combine like terms $- 4 z y$ and $- 4 y z$ .

Step 6.1:

Rearrange $z$ : $4 x^{2} + 4 x y + y^{2} - 4 y z - 4 z y - 16 x z + 16 z^{2}$ .

Step 6.2:

Combine $- 4 y z$ and $- 4 z y$ : $4 x^{2} + 4 x y + y^{2} - 8 y z - 16 x z + 16 z^{2}$ .

The final expanded form of $(2 x + y - 4 z)^{2}$ is $4 x^{2} + 4 x y - 8 y z - 16 x z + y^{2} + 16 z^{2}$ .

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., $a b = b a$ .
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., $a (b + c) = a b + a c$ .
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 $x^{2}$ .