Factor the trinomial [tex]$20 s^2+31 s-7$[/tex] completely.
Answer (2)
The trinomial 20 s 2 + 31 s − 7 can be factored into the product ( 5 s − 1 ) ( 4 s + 7 ) . This was achieved by finding two numbers that multiply to − 140 and add to 31 , rewriting the middle term, and then factoring by grouping. Finally, we combined the common binomial to arrive at the complete factorization.
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• Find two numbers whose product is 20 ⋅ ( − 7 ) = − 140 and whose sum is 31 . The numbers are 35 and − 4 .
• Rewrite the middle term: 20 s 2 + 31 s − 7 = 20 s 2 + 35 s − 4 s − 7 .
• Factor by grouping: 20 s 2 + 35 s − 4 s − 7 = 5 s ( 4 s + 7 ) − 1 ( 4 s + 7 ) .
• Factor out the common binomial factor: ( 5 s − 1 ) ( 4 s + 7 ) . The factored form is ( 5 s − 1 ) ( 4 s + 7 ) .
Explanation
Understanding the Problem We are asked to factor the trinomial 20 s 2 + 31 s − 7 completely. This means we want to express it as a product of two binomials.
Finding the Right Numbers To factor the trinomial, we look for two numbers whose product is equal to the product of the leading coefficient and the constant term, which is 20 × ( − 7 ) = − 140 , and whose sum is equal to the middle coefficient, which is 31 .
Identifying the Numbers The two numbers that satisfy these conditions are 35 and − 4 , since 35 × ( − 4 ) = − 140 and 35 + ( − 4 ) = 31 .
Rewriting the Middle Term Now we rewrite the middle term using these two numbers: 20 s 2 + 31 s − 7 = 20 s 2 + 35 s − 4 s − 7
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 20 s 2 + 35 s ) + ( − 4 s − 7 ) From the first group, we can factor out 5 s , and from the second group, we can factor out − 1 : 5 s ( 4 s + 7 ) − 1 ( 4 s + 7 )
Factoring out the Common Factor Now we factor out the common binomial factor ( 4 s + 7 ) from both terms: ( 5 s − 1 ) ( 4 s + 7 ) Thus, the trinomial is factored completely as ( 5 s − 1 ) ( 4 s + 7 ) .
Examples
Factoring trinomials is a fundamental skill in algebra and has many real-world applications. For example, if you are designing a rectangular garden and know the area can be represented by the trinomial 20 s 2 + 31 s − 7 , factoring it into ( 5 s − 1 ) ( 4 s + 7 ) helps you determine the possible dimensions (length and width) of the garden in terms of s . This is useful for planning the layout and optimizing the use of space.
