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Chapter 9: Problem 27

Write the expression in factored form. $$ 4 z^{2}-49 $$

Short Answer

Expert verified
Answer: The factored form of the expression \(4z^2 - 49\) is \((2z + 7)(2z - 7)\).

Step by step solution

01

Identify the difference of squares pattern

In the given expression \(4z^2 - 49\), notice that it has the form of a difference of squares. We can rewrite it as \((2z)^2 - (7)^2\). Here, \(a = 2z\) and \(b = 7\).
02

Apply the difference of squares formula

Now, let's apply the difference of squares formula:\((a^2 - b^2)=(a+b)(a-b)\). In our case, substitute \(a = 2z\) and \(b = 7\).
03

Factor the expression

Applying the formula with the given values, we get: \((2z+7)(2z-7)\). This is the factored form of the given expression, \(4z^2 - 49\).

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Key Concepts

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

Difference of Squares
The concept of the difference of squares is a key algebraic pattern that often appears in polynomial expressions. Recognizing this pattern simplifies the process of factoring. A difference of squares takes the form of \(a^2 - b^2\), where both terms are perfect squares, and they are separated by a subtraction sign.

To factor expressions using the difference of squares, you can follow these easy steps:
  • Identify the squares: Determine if the terms can be expressed as squares of other terms, such as \((2z)^2\) and \(7^2\).
  • Recognize the pattern: Ensure it fits the form \(a^2 - b^2\).
  • Use the formula: Apply the difference of squares formula, which is \((a+b)(a-b)\).
In the example \(4z^2 - 49\), by identifying it in the format \( (2z)^2 - (7)^2 \), you can easily factor it into \((2z+7)(2z-7)\).

This method transforms complex expressions into simpler ones, aiding further calculations.
Factored Form
The factored form of an expression is a product of simpler expressions that multiply to return the original polynomial. Simplifying an expression to its factored form can make solving equations and understanding the polynomial's properties more efficient.

When you factor a polynomial completely:
  • Break it down into components that cannot be factored further.
  • You use patterns or strategies, like the difference of squares, common factors, or grouping.
  • Check your work by multiplying the factors to ensure they return to the original expression.
For the expression \(4z^2 - 49\), factoring by using the difference of squares gives \((2z+7)(2z-7)\).
This means that multiplying \((2z+7)\) and \((2z-7)\) will result back to \(4z^2 - 49\).
By understanding the factored form, you simplify both the expression and the process of solving equations that contain it.
Algebraic Expressions
Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. They are the foundation of algebra and can represent a wide range of mathematical ideas and relationships.

Some key points to understand about algebraic expressions are:
  • Variables: Symbols, often letters, that represent unspecified numbers.
  • Constants: Fixed values that don't change.
  • Operations: Include addition, subtraction, multiplication, and division.
Expressions can be simple, like \(x + 2\), or complex, like \(4z^2 - 49\).
In the context of our example, \(4z^2 - 49\) is an algebraic expression that can be factored using the difference of squares.

By learning to manipulate and factor these expressions, you gain tools for solving algebraic equations efficiently and understanding the structure of mathematical relationships.

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Most popular questions from this chapter

Without solving them, say whether the equations in Problems \(49-56\) have two solutions, one solution, or no solution. Give a reason for your answer. $$ (x-3)^{2}=0 $$

Solve by (a) Completing the square (b) Using the quadratic formula $$ 9 x^{2}-6 x+1=0 $$

Write the quadratic function $$ y=8 x^{2}-2 x-15 $$ in the forms indicated.Give the values of all constants. $$ y=k x(v x+1)+w $$

Use what you know about the discriminant \(b^{2}-4 a c\) to decide what must be true about \(c\) in order for the quadratic equation \(3 x^{2}+2 x+c=0\) to have two different solutions.

Find the real numbers \(a\) and \(b\). \(15-25 i=3 a+5 b i\)
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