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Chapter 5: Problem 11

Find each product. $$ (4 a+5 b)^{2} $$

Short Answer

Expert verified
\[(4a + 5b)^2 = 16a^2 + 40ab + 25b^2\].

Step by step solution

01

Identify the binomial

The given expression is \( (4a + 5b)^2 \). This is a binomial raised to the power of 2.
02

Apply the Binomial Expansion Formula

Recall the formula for the square of a binomial: \[ (x + y)^2 = x^2 + 2xy + y^2 \]. In this case, \( x = 4a \) and \( y = 5b \).
03

Compute the Square of Each Term

Calculate \( x^2 \) and \( y^2 \): \( (4a)^2 = 16a^2 \) and \( (5b)^2 = 25b^2 \).
04

Compute the Double Product of the Two Terms

Calculate \( 2xy \): \[ 2 \times (4a) \times (5b) = 40ab \].
05

Combine All Terms

Combine all the computed terms: \[ (4a + 5b)^2 = 16a^2 + 40ab + 25b^2 \].

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Key Concepts

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

Polynomials
Polynomials are algebraic expressions made up of terms called monomials. A monomial is a product of numbers and variables raised to whole number powers. Polynomials can have one or more terms, and they are classified based on the number of terms they contain:
  • Monomial: A single term, like \(3x\).
  • Binomial: Two terms, like \(4a + 5b\).
  • Trinomial: Three terms, like \(x^2 - 3x + 2\).
Each term in a polynomial has a coefficient (the number) and a variable part (like \(x\), or \(a\) and \(b\) in our example). Understanding polynomials helps to simplify, add, subtract, and multiply these expressions, allowing you to solve complex mathematical problems easily.
Algebraic Expressions
An algebraic expression is a combination of constants (numbers), variables (letters), and operators (like \(+\), \(-\), \(\times\), and \(/\)). These expressions can represent real-world quantities and are essential for solving equations and inequalities.

There are a few key concepts to keep in mind when working with algebraic expressions:
  • Simplification: Combine like terms to make the expression easier to work with.
  • Substitution: Replace the variables with specific values to evaluate the expression.
  • Factoring: Write the expression as a product of its factors to simplify solving equations.
When dealing with the binomial \((4a + 5b)^2\), we need to use specific techniques like expansion to simplify it into a more manageable form.
Squared Binomials
A squared binomial is a special type of polynomial that results from multiplying a binomial by itself. This is often shown as \((x + y)^2\), which expands using the formula: \((x + y)^2 = x^2 + 2xy + y^2\). For our example, \((4a + 5b)^2\), we identify \(x = 4a\) and \(y = 5b\). Applying the formula, we get:
  • \(x^2 = (4a)^2 = 16a^2\)
  • \(y^2 = (5b)^2 = 25b^2\)
  • \(2xy = 2 \times 4a \times 5b = 40ab\)
Combining these terms: \((4a + 5b)^2 = 16a^2 + 40ab + 25b^2\). Squared binomials are a fundamental concept in algebra, and understanding how to expand them will help solve more complex polynomial equations and expressions.

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Most popular questions from this chapter

Perform each division. $$ \left(3 a^{2}-11 a+17\right) \div(2 a+6) $$

Evaluate. $$ 38 \div 10 $$

Write each product as a sum of terms. Write answers with positive exponents only. Simplify each term. See Section 1.8 $$ \frac{1}{4 y}\left(y^{4}+6 y^{2}+8\right) $$

Perform each division using the "long division" process. $$ \frac{6 r^{4}-11 r^{3}-r^{2}+16 r-8}{2 r-3} $$

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) See Example 1. $$ 9^{0} $$
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