Factor.
[tex]3 w^2+8 w+5[/tex]
Answer (2)
Find two numbers that multiply to 3 × 5 = 15 and add up to 8; these are 3 and 5.
Rewrite the middle term: 3 w 2 + 8 w + 5 = 3 w 2 + 3 w + 5 w + 5 .
Factor by grouping: 3 w ( w + 1 ) + 5 ( w + 1 ) .
Factor out the common factor: ( 3 w + 5 ) ( w + 1 ) . The factored form is ( 3 w + 5 ) ( w + 1 ) .
Explanation
Understanding the Problem We are given the quadratic expression 3 w 2 + 8 w + 5 and our goal is to factor it. Factoring means expressing the quadratic as a product of two binomials.
Finding the Right Numbers To factor the quadratic expression 3 w 2 + 8 w + 5 , we look for two numbers that multiply to 3 × 5 = 15 and add up to 8 .
Rewriting the Middle Term The two numbers that satisfy these conditions are 3 and 5 , since 3 × 5 = 15 and 3 + 5 = 8 . Now we rewrite the middle term using these numbers: 3 w 2 + 8 w + 5 = 3 w 2 + 3 w + 5 w + 5
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 3 w 2 + 3 w ) + ( 5 w + 5 ) Now, we factor out the greatest common factor from each group: 3 w ( w + 1 ) + 5 ( w + 1 )
Factoring out the Common Factor Now we see that ( w + 1 ) is a common factor in both terms. We factor out ( w + 1 ) : ( 3 w + 5 ) ( w + 1 ) So, the factored form of the quadratic expression is ( 3 w + 5 ) ( w + 1 ) .
Final Answer Therefore, the factored form of the given quadratic expression is ( 3 w + 5 ) ( w + 1 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Computer scientists use factoring in cryptography to secure data. Financial analysts use factoring to model and predict market trends. Factoring helps simplify complex problems and make them easier to solve.
To factor the quadratic expression 3 w 2 + 8 w + 5 , we find two numbers that multiply to 15 and add to 8 , which are 3 and 5 . By rewriting the middle term and then using factoring by grouping, we get the factored form ( 3 w + 5 ) ( w + 1 ) .
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