Problem 41
Question
For the following problems, find the products. $$ (x+1.3)^{2} $$
Step-by-Step Solution
Verified Answer
Answer: The expanded and simplified form of the expression \((x + 1.3)^2\) is \(x^2 + 2.6x + 1.69\).
1Step 1: Apply the binomial square formula
Using the binomial square formula \((a+b)^2 = a^2 + 2ab + b^2\), we'll substitute a = x and b = 1.3 into the formula and expand the expression.
$$
(x+1.3)^2 = (x)^{2} + 2(1.3)(x) + (1.3)^{2}
$$
2Step 2: Calculate the products
Compute the products in the expanded expression.
$$
(x)^{2} + 2(1.3)(x) + (1.3)^{2} = x^{2} + 2.6x + 1.69
$$
3Step 3: Write the final expression
The expanded and simplified expression is:
$$
(x+1.3)^2 = x^2 + 2.6x + 1.69
$$
Key Concepts
Binomial ExpansionPolynomial MultiplicationAlgebraic Expressions
Binomial Expansion
The concept of binomial expansion revolves around expanding expressions that involve two terms (a binomial) elevated to a power. The most elementary case is squaring a binomial, which is what our exercise deals with. The binomial square formula \( (a+b)^2 = a^2 + 2ab + b^2 \) expresses this expansion succinctly.
Using this formula, we see how any binomial such as \( (x+1.3)^2 \) can be expanded without actual multiplication. The key takeaway is that the coefficients of the expanded terms, in this case, always follow the sequence of 1, 2, and 1. This pattern is part of the larger Pascal’s Triangle used in more extensive binomial expansion. The formula demonstrates a shortcut by bypassing the need for polynomial multiplication, which is especially handy when working with higher powers.
Using this formula, we see how any binomial such as \( (x+1.3)^2 \) can be expanded without actual multiplication. The key takeaway is that the coefficients of the expanded terms, in this case, always follow the sequence of 1, 2, and 1. This pattern is part of the larger Pascal’s Triangle used in more extensive binomial expansion. The formula demonstrates a shortcut by bypassing the need for polynomial multiplication, which is especially handy when working with higher powers.
Polynomial Multiplication
Polynomial multiplication is the process of multiplying algebraic expressions that consist of variables raised to whole number exponents. In our example, \( (x+1.3)^2 \) could also be computed using the distributive property, which in essence is the foundation for polynomial multiplication.
The distributive property allows us to multiply each term in one polynomial by each term in the other. For squaring a binomial, it means multiplying \( x \) by \( x \) and \( 1.3 \) by \( 1.3 \), but also cross-multiplying \( x \) by \( 1.3 \) and then doubling the result because it will appear twice (once for each order of multiplication). This manual method becomes tedious for higher powers but remains a crucial concept in understanding the interaction between terms when they multiply, laying a foundation for algebraic manipulation and simplification.
The distributive property allows us to multiply each term in one polynomial by each term in the other. For squaring a binomial, it means multiplying \( x \) by \( x \) and \( 1.3 \) by \( 1.3 \), but also cross-multiplying \( x \) by \( 1.3 \) and then doubling the result because it will appear twice (once for each order of multiplication). This manual method becomes tedious for higher powers but remains a crucial concept in understanding the interaction between terms when they multiply, laying a foundation for algebraic manipulation and simplification.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain ordinary numbers, variables (like \( x \) or \( y \) ), and operations (like addition, subtraction, multiplication, and division). They are the cornerstone of algebra and are used to represent relationships and to solve problems.
The final expanded form of \( (x+1.3)^2 \) is \( x^2 + 2.6x + 1.69 \), which is an algebraic expression. This particular expression shows the outcome of applying specific algebraic rules (like binomial expansion) to a given problem. It conveys much information, such as what happens to \( x \) when it is squared and then combined with another term. Grasping algebraic expressions is vital, as they are instrumental in forming equations that model real world scenarios, from physics to finance.
The final expanded form of \( (x+1.3)^2 \) is \( x^2 + 2.6x + 1.69 \), which is an algebraic expression. This particular expression shows the outcome of applying specific algebraic rules (like binomial expansion) to a given problem. It conveys much information, such as what happens to \( x \) when it is squared and then combined with another term. Grasping algebraic expressions is vital, as they are instrumental in forming equations that model real world scenarios, from physics to finance.
Other exercises in this chapter
Problem 40
For the following problems, perform the multiplications and combine any like terms. $$ -3(b+8) $$
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For the following problems, list, if any should appear, the common factors in the expressions. $$ 6 x^{2} y^{3}+18 x^{2} $$
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Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ y^{2}+3=2 y-6 $$
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For the following problems, simplify each of the algebraic expressions. $$ 18 x^{2} y-14 x^{2} y-20 x^{2} y $$
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