Problem 8 - Precalculus: Mathematics for Calculus

Question

Yes or No? II No, give a reason. (a) Is the expression \((x+5)^{2}\) equal to \(x^{2}+25 ?\) (b) When you expand \((x+a)^{2},\) where \(a \neq 0,\) do you get three terms? (c) Is the expression \((x+5)(x-5)\) equal to \(x^{2}-25 ?\) (d) When you expand \((x+a)(x-a),\) where \(a \neq 0,\) do you get two terms?

Step-by-Step Solution

Answer

(a) No, they are not equal. (b) Yes, it expands to three terms. (c) Yes, they are equal. (d) Yes, it results in two terms.

Step 1: Evaluate Expression (a)

To determine if \((x+5)^2\) is equal to \(x^2 + 25\), we need to expand \((x+5)^2\). Expanding, we get: \((x+5)(x+5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25\). Since \(x^2 + 10x + 25\) is not equal to \(x^2 + 25\), the answer is No.

Step 2: Evaluate Expression (b)

To find out if expanding \((x+a)^2\) results in three terms, let's expand it: \((x+a)(x+a) = x^2 + ax + ax + a^2 = x^2 + 2ax + a^2\). The result has three terms (\(x^2\), \(2ax\), and \(a^2\)), so the answer is Yes.

Step 3: Evaluate Expression (c)

To determine if \((x+5)(x-5)\) is equal to \(x^2 - 25\), expand \((x+5)(x-5)\) using the difference of squares formula: \((x+5)(x-5) = x^2 - 5^2 = x^2 - 25\). The expressions are equal, so the answer is Yes.

Step 4: Evaluate Expression (d)

To find out if expanding \((x+a)(x-a)\) results in two terms, use the difference of squares formula: \((x+a)(x-a) = x^2 - a^2\), which has only two terms. Therefore, the answer is Yes.

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. These expressions play a crucial role in algebra and appear in various mathematical problems. A polynomial is often written in the following form:

• Each term in a polynomial has a power of the variable, known as the degree, which is a non-negative integer.
• For example, in the expression \(x^2 + 3x + 2\), the degrees are 2, 1, and 0, respectively.
• Polynomials can be classified based on the number of terms: monomials (one term), binomials (two terms), and trinomials (three terms).

Understanding polynomials is essential for expanding expressions like \((x+5)^2\), which results in terms of different degrees, demonstrating how terms combine.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. These expressions play a foundational role in algebra and form the basis for solving algebraic equations.

• They can include one or more terms and can involve operations such as addition (\(+\)), subtraction (\(-\)), multiplication (\(\times\)), and division (\(\div\)).
• For example, \((x+a)^2\) expands to \(x^2 + 2ax + a^2\), showcasing operations of multiplication and addition.
• Each term in an algebraic expression can be a constant or involve variables raised to a power.

The goal of manipulating these expressions, like expanding, is to simplify or solve for unknown values.

Difference of Squares

The difference of squares is a special algebraic formula used to simplify products of certain binomials. This formula states that when you have two terms, each raised to the square, and subtracted, they can be expressed as: \[ (x+a)(x-a) = x^2 - a^2\]

• This pattern works because the middle terms cancel each other out, leaving only the square of the first term minus the square of the second term.
• Recognizing this pattern is helpful for quickly factoring or expanding expressions such as \((x+5)(x-5)\), which simplifies to \(x^2 - 25\).
• The formula is often used in algebra for simplifying expressions and solving equations more efficiently.

Understanding the difference of squares can streamline solving certain algebraic problems.