Factor the following expression:
9) $5 p^2-p-18$
$(5 p+9)(p-2)$
Answer (1)
Expand the factored form ( 5 p + 9 ) ( p − 2 ) using the distributive property.
Multiply each term: 5 p 2 − 10 p + 9 p − 18 .
Combine like terms: 5 p 2 − p − 18 .
The expanded form matches the original expression, so the factorization is correct: ( 5 p + 9 ) ( p − 2 ) .
Explanation
Problem Analysis We are given the quadratic expression 5 p 2 − p − 18 and its proposed factorization ( 5 p + 9 ) ( p − 2 ) . Our task is to verify whether the factorization is correct. To do this, we will expand the factored form and see if it matches the original quadratic expression.
Expanding the Factored Form To expand the factored form ( 5 p + 9 ) ( p − 2 ) , we use the distributive property (also known as the FOIL method):
( 5 p + 9 ) ( p − 2 ) = 5 p ( p ) + 5 p ( − 2 ) + 9 ( p ) + 9 ( − 2 )
Performing Multiplications Now, we perform the multiplications:
5 p ( p ) = 5 p 2 5 p ( − 2 ) = − 10 p 9 ( p ) = 9 p 9 ( − 2 ) = − 18
So, the expanded expression is:
5 p 2 − 10 p + 9 p − 18
Combining Like Terms Next, we combine the like terms (the terms with the same power of p ):
− 10 p + 9 p = − p
So, the simplified expanded expression is:
5 p 2 − p − 18
Conclusion Finally, we compare the simplified expanded expression 5 p 2 − p − 18 with the original quadratic expression 5 p 2 − p − 18 . Since they are identical, the factorization is correct.
Therefore, the factorization of 5 p 2 − p − 18 is indeed ( 5 p + 9 ) ( p − 2 ) .
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 an area represented by the quadratic expression 5 p 2 − p − 18 . Factoring this expression into ( 5 p + 9 ) ( p − 2 ) helps you determine the possible dimensions (length and width) of the garden. This skill is crucial for problem-solving in many fields.
