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Chapter 4: Problem 40

Find each product. $$ p(3 p+7)(3 p-7) $$

Short Answer

Expert verified
9p^2 - 49

Step by step solution

01

- Identify the expression structure

Notice that the expression \( (3p + 7)(3p - 7) \) fits the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 3p\) and \(b = 7\).
02

- Apply the difference of squares formula

Using the formula for the difference of squares, rewrite the expression: \((3p + 7)(3p - 7) = (3p)^2 - 7^2\).
03

- Simplify the squares

Calculate the squares of both terms: \( (3p)^2 = 9p^2 \) and \( 7^2 = 49 \).
04

- Subtract the constants

Subtract the squared terms: \( 9p^2 - 49 \).

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Key Concepts

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

Factoring Techniques
Factoring is a key concept in algebra that helps simplify expressions and solve equations. One important technique is the difference of squares. This occurs when an expression is in the form \(a^2 - b^2\). For example, an expression like \((3p + 7)(3p - 7)\) can be factored as a difference of squares. Here鈥檚 how it works:
  • Rewrite the expression in the form \(a^2 - b^2\).
  • If the expression looks like \((a + b)(a - b)\), identify \(\textbf{a}\) and \(\textbf{b}\).
  • In our example, \(a = 3p\) and \(\textbf{b} = 7\).
  • Applying the formula gives us \(a^2 - b^2 = (3p)^2 - 7^2\).
  • Simplify as \((3p)^2 = 9p^2\) and \(\textbf{7^2 = 49}\).
  • Thus, \((3p + 7)(3p - 7)\) simplifies to \(9p^2 - 49\).
This formula is very powerful for polynomials that have a clear pattern of squared terms being subtracted. Practicing this technique will make it easier to deal with more complex algebraic expressions.
Algebraic Expressions
Algebraic expressions combine numbers, variables, and mathematical operations to describe relationships and quantities. An example is \(3p + 7\) or \(3p - 7\). These form the basic building blocks in algebra.
  • Terms: Individual parts of an expression separated by \( + \textbackslash{} \textbackslash{} - \). In \(3p + 7\), \(\textbf{3p}\) and \(\textbf{7}\) are terms.
  • Factors: Numbers or variables that multiply together to form a term. In the term \(3p\), \(\textbf{3}\) and \(p\) are factors.
  • Coefficients: The numerical part of a term. In \(\textbf{3p}\), \(\textbf{3}\) is the coefficient.
  • Constants: Numbers without variables. In \(\textbf{7}\), \( \textbf{7} \) is a constant.
Understanding and manipulating algebraic expressions is crucial because it allows us to model real-world situations and solve problems efficiently. Breaking down expressions into simpler parts by factoring, like with the difference of squares, is an essential skill.
Polynomials
Polynomials are expressions consisting of variables and coefficients, connected by addition, subtraction, and multiplication operations. They look like \(9p^2 - 49\). More formally, a polynomial in one variable \(x\) looks like: \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are coefficients and \(n\) is a non-negative integer.
  • Monomials: Polynomials with a single term like \(3p\) or \(-5x^2\).
  • Binomials: Polynomials with two terms such as \(9p^2 - 49\) (after using the difference of squares).
  • Trinomials: Polynomials with three terms.
  • Degrees: The highest power of the variable in the polynomial. For \(9p^2 - 49\), the degree is 2.
Polynomials are used to describe a wide range of phenomena in science, engineering, and everyday life. Simplifying and factoring polynomials, like we did with the difference of squares, is a foundational skill in algebra that helps us to understand and solve more complex problems.

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