Problem 40 Expand and multiply. $$(x+7)^{... [FREE SOLUTION]

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Prealgebra

Charles P. McKeague

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 9: Problem 40

Expand and multiply. $$(x+7)^{2}$$

Short Answer

Expert verified

$(x+7)^2 = x^2 + 14x + 49$.

Step by step solution

Recognize the Expression Format

The expression $(x + 7)^2$ is a binomial square. A binomial square is an expression where a binomial term is multiplied by itself.

Apply the Binomial Theorem for Squaring

Use the formula for squaring a binomial: $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = x$ and $b = 7$.

Calculate $a^2$

Square $a$, which is $x$: \[x^2 = x \times x = x^2\]

Calculate $2ab$

Compute the product $2ab$: \[2ab = 2 \times x \times 7 = 14x\]

Calculate $b^2$

Square $b$, which is 7:\[b^2 = 7 \times 7 = 49\]

Combine the Results

Add all the calculated parts together to expand the expression: \[x^2 + 14x + 49\]

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Key Concepts

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

Squaring Binomials

When faced with an expression such as $(x + 7)^2$, you're dealing with a classic example of squaring binomials. A binomial is simply an algebraic expression containing two terms, typically in the form $(a + b)$. Squaring a binomial involves multiplying the binomial by itself.

We have $(a + b)^2$ which means $(a + b) \times (a + b)$.
This is where the binomial theorem comes in handy, providing a formula to simplify the process.
The formula states that $(a + b)^2 = a^2 + 2ab + b^2$.

By recognizing this pattern, we avoid the more cumbersome process of direct multiplication by simplifying each term with calculated precision.

Algebraic Expressions

An algebraic expression like $(x + 7)^2$ can seem intimidating at first glance, but understanding its components is key to mastering algebra. At its core, an algebraic expression is a mathematical phrase that includes numbers, variables, and operators, such as addition or multiplication.

The expression $(x + 7)^2$ features a variable $x$ and a constant 7.
Variables like $x$ represent unknown values and can change, whereas constants are fixed numbers.
Algebraic expressions depend on the rules of arithmetic and algebra to combine these elements effectively.

Deciphering algebraic expressions involves identifying these parts and using appropriate operations to expand or simplify as needed.

Polynomials

Expanding $(x + 7)^2$ transforms the expression into a polynomial, a fundamental concept encompassing expressions with one or more terms. By performing the steps outlined in our solution, we go from a simple binomial to a trinomial polynomial.

The expanded expression $x^2 + 14x + 49$ consists of three terms.
Each term within the polynomial represents a degree, related to the power of the variable it contains.
The term $x^2$ is of degree 2, $14x$ is of degree 1, and $49$ is a constant term of degree 0.

Polynomials are versatile, playing a crucial role in various areas of mathematics and are foundational to algebraic reasoning. Understanding polynomials begins with breaking down complex expressions into simpler, manageable parts.

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91影视

Expand and multiply. $$(x+7)^{2}$$

Short Answer

Step by step solution

Recognize the Expression Format

Apply the Binomial Theorem for Squaring

Calculate \(a^2\)

Calculate \(2ab\)

Calculate \(b^2\)

Combine the Results

Key Concepts

Squaring Binomials

Algebraic Expressions

Polynomials

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