25. [tex]y^2+\frac{b^2-1}{b} y-1=[/tex]
a) [tex](y-b)\left(y-\frac{1}{b}\right)[/tex]
b) [tex](y+b)\left(y+\frac{1}{b}\right)[/tex]
c) [tex]\left(y-\frac{1}{b}\right)(y+b)[/tex]
d) [tex]\left(y+\frac{1}{b}\right)(y-b)[/tex]
Answer (1)
Expand each of the given options.
Compare the expanded form of each option with the given quadratic expression.
Option c) matches the given quadratic expression.
The correct factorization is ( y − b 1 ) ( y + b ) .
Explanation
Problem Analysis We are given the quadratic expression y 2 + b b 2 − 1 y − 1 and asked to find its correct factorization from the given options.
Expanding the Options Let's expand each of the options to see which one matches the given quadratic expression. Option a) ( y − b ) ( y − b 1 ) = y 2 − b 1 y − b y + 1 = y 2 − ( b + b 1 ) y + 1 = y 2 − ( b b 2 + 1 ) y + 1 Option b) ( y + b ) ( y + b 1 ) = y 2 + b 1 y + b y + 1 = y 2 + ( b + b 1 ) y + 1 = y 2 + ( b b 2 + 1 ) y + 1 Option c) ( y − b 1 ) ( y + b ) = y 2 + b y − b 1 y − 1 = y 2 + ( b − b 1 ) y − 1 = y 2 + ( b b 2 − 1 ) y − 1 Option d) ( y + b 1 ) ( y − b ) = y 2 − b y + b 1 y − 1 = y 2 + ( b 1 − b ) y − 1 = y 2 − ( b − b 1 ) y − 1 = y 2 − ( b b 2 − 1 ) y − 1
Identifying the Correct Factorization Comparing the expanded forms with the given quadratic expression y 2 + b b 2 − 1 y − 1 , we see that option c) matches the given expression.
Final Answer Therefore, the correct factorization is ( y − b 1 ) ( y + b ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in various real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to optimize algorithms. In simple terms, imagine you are designing a rectangular garden with a specific area and you need to determine the possible dimensions (length and width). Factoring the quadratic equation representing the area can help you find these dimensions efficiently.
