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Chapter 6: Problem 3

Expand and simplify each expression. $$ (x+2.5)^{2} $$

Short Answer

Expert verified
The expanded and simplified form is \( x^2 + 5x + 6.25 \).

Step by step solution

01

Understand the Problem

The task is to expand and simplify the expression \( (x + 2.5)^2 \). This means applying the binomial theorem or distributive property.
02

Apply the Binomial Theorem

The binomial theorem for \( (a + b)^2 \) states that \[ (a + b)^2 = a^2 + 2ab + b^2. \] In this case, \( a = x \) and \( b = 2.5 \).
03

Square the First Term

Square the first term \( a = x \): \[ x^2 \]
04

Multiply the Two Terms and Then Multiply by 2

Multiply \( x \) and \( 2.5 \) together, then multiply the result by 2: \[ 2 \times (x \times 2.5) = 5x \]
05

Square the Second Term

Square the second term \( b = 2.5 \): \[ (2.5)^2 = 6.25 \]
06

Combine All Parts

Combine all the parts calculated in the previous steps: \[ x^2 + 5x + 6.25 \]

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Key Concepts

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

binomial theorem
The binomial theorem is a fundamental concept in algebra that helps expand expressions raised to a power.
For any binomial expression \( (a + b)^n \), the theorem provides a formula to expand it into a sum involving terms of the form \ a^k \ and \ b^{n-k} \.
In our example, we used the binomial theorem for \( (x + 2.5)^2 \).
The general form for squaring is: \[ (a + b)^2 = a^2 + 2ab + b^2. \]
This formula helps us break the expression into manageable parts: \ x^2 \, \ 2x \times 2.5 \, and \ (2.5)^2 \.
polynomials
Polynomials are expressions involving variables and coefficients, structured as a sum of terms.
Each term is the product of a constant and a variable raised to a non-negative integer power.
The given exercise deals with expanding a polynomial in its simplified form.
Here, we started with the binomial \( (x + 2.5)^2 \) and ended with the polynomial \ x^2 + 5x + 6.25 \, which is a quadratic polynomial.
Quadratic polynomials have degrees of 2, with the general form: \[ ax^2 + bx + c \].
In the final polynomial \ x^2 + 5x + 6.25 \, we identified each part:
  • \ a = 1, representing the coefficient of \ x^2 \.
  • \

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Most popular questions from this chapter

Rewrite each expression using a single base and a single exponent. $$ a^{12} \cdot\left(a^{2}\right)^{-7} $$

Write each expression as the product of binomials. $$ 16-25 x^{2} $$

Use the distributive property to expand each expression. $$ \frac{1}{2} x(x-2) $$

Expand each expression. Simplify your expansion if possible. \((4 x+1)^{2}\)

Expand each expression. Simplify your results by combining like terms. \(2(x+y)+x(3+y)(x+2)\)
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