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Chapter 6: Problem 27

Use a pattern to factor. Check. Identify any prime polynomials. $$ 4 f^{2}-25 $$

Short Answer

Expert verified
The polynomial 4f^{2}-25 factors as (2f + 5)(2f - 5) and the factors are prime.

Step by step solution

01

Identify the structure of the polynomial

The given expression is 4f^{2}-25. Note that both terms are perfect squares: 4f^{2} = (2f)^{2} 25 = 5^{2}. Thus, this polynomial is a difference of squares.
02

Apply the difference of squares formula

Recall the difference of squares formula: a^{2} - b^{2} = (a + b)(a 鈥� b). Here, a = 2f and b = 5. Thus: 4f^{2} - 25 = (2f)^{2} - 5^{2} = (2f + 5)(2f - 5).
03

Check if the factors are prime

The factors (2f + 5) and (2f - 5) are both linear polynomials and cannot be factored further. Therefore, they are prime polynomials.
04

Verify the solution

Multiply the factors to check your work: (2f + 5)(2f - 5). Using the distributive property (FOIL method): = 2f(2f) - 2f(5) + 5(2f) - 5(5) = 4f^{2} - 10f + 10f - 25. The middle terms cancel out: = 4f^{2} - 25. This confirms the factorization is correct.

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Key Concepts

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

Difference of Squares
The difference of squares is a special algebraic expression that involves the subtraction of two perfect squares. It can be factored into the product of two binomials. The general form of a difference of squares is given by: \(a^{2} - b^{2} = (a + b)(a - b)\). In this pattern, one term is subtracted from another, and each term is a perfect square.
For example, in the polynomial \(4f^{2} - 25\), we see two perfect squares: \((2f)^{2}\) and \(5^{2}\). By identifying these, we can apply the difference of squares formula to factor the expression.
Prime Polynomials
Prime polynomials are polynomials that cannot be factored further into polynomials of lower degree with integer coefficients. In other words, they are the basic building blocks of polynomials, much like prime numbers in the realm of integers.
After factoring \(4f^{2} - 25\) into \((2f + 5)(2f - 5)\), we are left with two binomials. Each binomial is a linear polynomial, meaning they are degree 1 polynomials that cannot be further simplified. Therefore, \(2f + 5\) and \(2f - 5\) are considered prime polynomials.
FOIL Method
The FOIL method is a handy technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to the four products you must sum to find the product of the binomials.
For example, to verify the factorization of \(4f^{2} - 25\), we can use the FOIL method on \((2f + 5)(2f - 5)\):
  • First: \(2f \times 2f = 4f^{2}\)
  • Outer: \(2f \times (-5) = -10f\)
  • Inner: \(5 \times 2f = 10f\)
  • Last: \(5 \times (-5) = -25\)
Adding these products together: \(4f^{2} - 10f + 10f - 25\). Notice that \(-10f\) and \(10f\) cancel each other out, leaving \(4f^{2} - 25\), which confirms the factorization is correct.
Perfect Squares
A perfect square is a number or expression that can be written as the square of something else. In terms of polynomials, a term like \(4f^{2}\) is a perfect square because it can be written as \((2f)^{2}\), and \(25\) is a perfect square because it can be written as \(5^{2}\).
Recognizing perfect squares is crucial for factoring polynomials, especially when applying the difference of squares formula.
Understanding perfect squares and how they contribute to polynomial structure helps in identifying factorization opportunities, making problems easier to solve.

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