Factor this polynomial:
$-3 x^2-16 x-5$
A. $(3 x-1)(x-5)$
B. $-1(3 x+1)(x+5)$
C. $(-3 x+1)(x+5)$
D. $(3 x+1)(x-5)$
Answer (2)
Factor out a -1: − ( 3 x 2 + 16 x + 5 ) .
Find two numbers that multiply to 15 and add to 16 (1 and 15).
Rewrite and factor by grouping: 3 x 2 + 15 x + x + 5 = ( 3 x + 1 ) ( x + 5 ) .
The final factorization is − 1 ( 3 x + 1 ) ( x + 5 ) .
Explanation
Understanding the Problem We are given the quadratic polynomial − 3 x 2 − 16 x − 5 and asked to factor it. We are also given four possible factorizations and must choose the correct one.
Factoring out -1 First, factor out a -1 from the polynomial: − 3 x 2 − 16 x − 5 = − ( 3 x 2 + 16 x + 5 )
Finding the right numbers Now, we need to factor the quadratic 3 x 2 + 16 x + 5 . We are looking for two numbers that multiply to 3 × 5 = 15 and add to 16. These numbers are 1 and 15.
Rewriting the middle term Rewrite the middle term using the numbers we found: 3 x 2 + 16 x + 5 = 3 x 2 + 15 x + x + 5
Factoring by grouping Factor by grouping: 3 x 2 + 15 x + x + 5 = 3 x ( x + 5 ) + 1 ( x + 5 )
Factoring out the common factor Factor out the common factor ( x + 5 ) : 3 x ( x + 5 ) + 1 ( x + 5 ) = ( 3 x + 1 ) ( x + 5 )
Final Factorization Therefore, the factorization of the original polynomial is − 3 x 2 − 16 x − 5 = − ( 3 x + 1 ) ( x + 5 )
Selecting the correct option Comparing our result with the given options, we see that option B matches our factorization.
Examples
Factoring polynomials 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 market behavior, and computer scientists use it to design efficient algorithms. Imagine you are designing a bridge and need to ensure it can withstand certain loads. By expressing the load distribution as a polynomial and factoring it, you can identify critical points where the stress is highest, allowing you to reinforce those areas and ensure the bridge's safety. This process helps in optimizing designs and preventing potential failures.
To factor the polynomial − 3 x 2 − 16 x − 5 , we first factor out − 1 , resulting in − 1 ( 3 x 2 + 16 x + 5 ) . The quadratic 3 x 2 + 16 x + 5 factors to ( 3 x + 1 ) ( x + 5 ) , leading us to the final factorization of − 1 ( 3 x + 1 ) ( x + 5 ) , which matches option B.
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