Factor.
$5 z^2+26 z+5$
Answer (1)
Find two numbers whose product equals a c and whose sum equals b .
Rewrite the middle term using these two numbers.
Factor by grouping.
Factor out the common factor to obtain the factored form: ( 5 z + 1 ) ( z + 5 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression 5 z 2 + 26 z + 5 . This is a quadratic trinomial of the form a z 2 + b z + c , where a = 5 , b = 26 , and c = 5 .
Finding the Right Numbers To factor this quadratic, we look for two numbers whose product is a c = 5 × 5 = 25 and whose sum is b = 26 .
Rewriting the Middle Term The two numbers are 1 and 25, since 1 × 25 = 25 and 1 + 25 = 26 . Now we rewrite the middle term using these two numbers: 5 z 2 + 26 z + 5 = 5 z 2 + z + 25 z + 5
Factoring by Grouping Next, we factor by grouping: 5 z 2 + z + 25 z + 5 = z ( 5 z + 1 ) + 5 ( 5 z + 1 )
Factoring out the Common Factor Finally, we factor out the common factor ( 5 z + 1 ) : z ( 5 z + 1 ) + 5 ( 5 z + 1 ) = ( 5 z + 1 ) ( z + 5 ) Thus, the factored form of the quadratic expression is ( 5 z + 1 ) ( z + 5 ) .
Final Answer Therefore, the factored form of the given quadratic expression is ( 5 z + 1 ) ( z + 5 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many 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 design efficient algorithms. Factoring also helps in solving quadratic equations, which can model projectile motion, optimization problems, and various other phenomena.
