Factor the trinomial completely.
[tex]12 w^2+11 w-15[/tex]
Answer (2)
• Find two numbers whose product is 12 × − 15 = − 180 and whose sum is 11 . The numbers are 20 and − 9 .
• Rewrite the middle term: 12 w 2 + 11 w − 15 = 12 w 2 + 20 w − 9 w − 15 .
• Factor by grouping: ( 12 w 2 + 20 w ) + ( − 9 w − 15 ) = 4 w ( 3 w + 5 ) − 3 ( 3 w + 5 ) .
• Factor out the common binomial: ( 3 w + 5 ) ( 4 w − 3 ) . The factored form is ( 3 w + 5 ) ( 4 w − 3 ) .
Explanation
Understanding the Problem We are given the quadratic trinomial 12 w 2 + 11 w − 15 . Our goal is to factor this trinomial completely. Factoring involves expressing the trinomial as a product of two binomials.
Finding the Right Numbers To factor the trinomial 12 w 2 + 11 w − 15 , we need to find two numbers whose product is equal to the product of the leading coefficient (12) and the constant term (-15), which is 12 × − 15 = − 180 , and whose sum is equal to the middle coefficient, which is 11.
Identifying the Numbers Let's find these two numbers. We are looking for two numbers that multiply to -180 and add up to 11. After some consideration, we find that the numbers are 20 and -9, since 20 × − 9 = − 180 and 20 + ( − 9 ) = 11 .
Rewriting the Middle Term Now we rewrite the middle term using these two numbers: 12 w 2 + 11 w − 15 = 12 w 2 + 20 w − 9 w − 15 .
Grouping the Terms Next, we factor by grouping. We group the first two terms and the last two terms: ( 12 w 2 + 20 w ) + ( − 9 w − 15 ) .
Factoring out the GCF We factor out the greatest common factor (GCF) from each group. The GCF of 12 w 2 and 20 w is 4 w , so we have 4 w ( 3 w + 5 ) . The GCF of − 9 w and − 15 is − 3 , so we have − 3 ( 3 w + 5 ) . Thus, we have 4 w ( 3 w + 5 ) − 3 ( 3 w + 5 ) .
Factoring out the Common Binomial Now we factor out the common binomial factor ( 3 w + 5 ) from the expression 4 w ( 3 w + 5 ) − 3 ( 3 w + 5 ) . This gives us ( 3 w + 5 ) ( 4 w − 3 ) .
Final Factorization Therefore, the factored form of the trinomial 12 w 2 + 11 w − 15 is ( 3 w + 5 ) ( 4 w − 3 ) .
Examples
Factoring trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and analyze stress. Architects use factoring to create blueprints and calculate dimensions. Financial analysts use factoring to model investment portfolios and predict market trends. By mastering factoring, you'll be equipped to solve a wide range of problems in various fields.
To factor 12 w 2 + 11 w − 15 , we rewrite it by finding two numbers that sum to 11 and multiply to -180, which are 20 and -9. We then group and factor the expression to get ( 3 w + 5 ) ( 4 w − 3 ) . The final factored form is ( 3 w + 5 ) ( 4 w − 3 ) .
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