Problem 315 Square each binomial using the B... [FREE SOLUTION]

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Chapter 6: Problem 315

Square each binomial using the Binomial Squares Pattern. $$(2 y-3 z)^{2}$$

Short Answer

Expert verified

(2y - 3z)^2 = 4y^2 - 12yz + 9z^2

Step by step solution

Recall the Binomial Squares Pattern

The Binomial Squares Pattern states \[ (a - b)^2 = a^2 - 2ab + b^2 \]

Identify the Terms

Identify the terms in the given binomial. Here, the binomial is $ (2y - 3z) $.Let $ a = 2y $ and $ b = 3z $.

Square the First Term

Square the first term $ a = 2y $: \[ (2y)^2 = 4y^2 \]

Square the Second Term

Square the second term $ b = 3z $: \[ (3z)^2 = 9z^2 \]

Apply the Middle Term

Apply the middle term which is $ -2ab $: \[ -2(2y)(3z) = -12yz \]

Combine All Terms

Combine all the terms together: \[ (2y - 3z)^2 = 4y^2 - 12yz + 9z^2 \]

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Key Concepts

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

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra and are essential for solving various mathematical problems.
For example, in the expression $2y - 3z$, $2y$ and $3z$ are terms with variables $y$ and $z$.
Here, $2$ and $-3$ are coefficients. Understanding these components helps to simplify and manipulate expressions effectively.

Binomials

A binomial is a type of algebraic expression that has exactly two terms. These terms are usually separated by a plus (+) or minus (-) sign.
For example, in the exercises above, $2y - 3z$ is a binomial.
Binomials can be manipulated using various algebraic techniques, such as squaring.

Squaring Techniques

Squaring a binomial means multiplying the binomial by itself. The Binomial Squares Pattern provides a simple way to square binomials. According to this pattern: $((a \pm b)^{2} = a^{2} \pm 2ab + b^{2})$.
You start by identifying $a$ and $b$ in your binomial: for $2y - 3z$, $a = 2y$ and $b = 3z$.
Next, square each term: $a^{2}$ becomes $ (2y)^{2} = 4y^{2}$ and $(3z)^{2}$ = 9z^{2}. Finally, calculate the middle term: $-2ab = -12yz$. Combine all terms to get: $ (2y - 3z)^{2} = 4y^{2} - 12yz + 9z^{2}$.

Polynomials

A polynomial is an algebraic expression made up of one or more terms. These terms can include variables, coefficients, and exponents.
Polynomials can have many forms, such as monomials (one term), binomials (two terms), or even higher-order polynomials with multiple terms.
For instance, the result $ 4y^{2} - 12yz + 9z^{2} $, is a polynomial that came from squaring the binomial $2y - 3z$.
Each term in this polynomial represents a component of the original binomial, showing the power of algebraic manipulation.

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91影视

Square each binomial using the Binomial Squares Pattern. $$(2 y-3 z)^{2}$$

Short Answer

Step by step solution

Recall the Binomial Squares Pattern

Identify the Terms

Square the First Term

Square the Second Term

Apply the Middle Term

Combine All Terms

Key Concepts

Algebraic Expressions

Binomials

Squaring Techniques

Polynomials

One App. One Place for Learning.

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