Completely factor the trinomial, if possible.
[tex]$15 x^2+32 x-7$[/tex]
Answer (2)
Find two numbers whose product is 15 × − 7 = − 105 and whose sum is 32 , which are 35 and − 3 .
Rewrite the middle term: 15 x 2 + 35 x − 3 x − 7 .
Factor by grouping: 5 x ( 3 x + 7 ) − 1 ( 3 x + 7 ) .
Factor out the common binomial: ( 3 x + 7 ) ( 5 x − 1 ) . The factored form is ( 3 x + 7 ) ( 5 x − 1 ) .
Explanation
Understanding the Problem We are given the trinomial 15 x 2 + 32 x − 7 . Our goal is to factor this trinomial completely, if possible. Factoring involves expressing the trinomial as a product of two binomials.
Finding the Right Numbers We will use the factoring by grouping method. This method involves finding two numbers whose product is equal to the product of the leading coefficient and the constant term, and whose sum is equal to the middle coefficient. In this case, we need two numbers whose product is 15 × − 7 = − 105 and whose sum is 32 .
Rewriting the Middle Term The two numbers that satisfy these conditions are 35 and − 3 , since 35 × − 3 = − 105 and 35 + ( − 3 ) = 32 . Now we rewrite the middle term using these two numbers: 15 x 2 + 35 x − 3 x − 7 .
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 15 x 2 + 35 x ) + ( − 3 x − 7 ) . Now we factor out the greatest common factor from each group. From the first group, we can factor out 5 x , which gives us 5 x ( 3 x + 7 ) . From the second group, we can factor out − 1 , which gives us − 1 ( 3 x + 7 ) . So we have 5 x ( 3 x + 7 ) − 1 ( 3 x + 7 ) .
Factoring out the Common Binomial Now we factor out the common binomial factor, which is ( 3 x + 7 ) . This gives us ( 3 x + 7 ) ( 5 x − 1 ) . Therefore, the factored form of the trinomial is ( 3 x + 7 ) ( 5 x − 1 ) .
Final Answer The completely factored form of the trinomial 15 x 2 + 32 x − 7 is ( 3 x + 7 ) ( 5 x − 1 ) .
Examples
Factoring trinomials is useful in many real-world applications, such as optimizing areas and volumes in construction and engineering. For example, if you are designing a rectangular garden with an area represented by the trinomial 15 x 2 + 32 x − 7 , factoring it into ( 3 x + 7 ) ( 5 x − 1 ) helps you determine the possible dimensions of the garden in terms of x . This allows you to plan the layout efficiently and make the best use of available space.
The trinomial 15 x 2 + 32 x − 7 can be completely factored as ( 3 x + 7 ) ( 5 x − 1 ) . This is achieved by finding two numbers that multiply to − 105 (the product of the leading coefficient and the constant term) and add to 32 . Through the process of grouping and factoring, we arrive at the final factored form.
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