Factor the given polynomial.
$3 x^2+7 x-20$
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A. $3 x^2+7 x-20=$ $\square$
B. The polynomial is prime.
Answer (1)
Find two numbers that multiply to 3 × − 20 = − 60 and add up to 7 , which are 12 and − 5 .
Rewrite the middle term: 3 x 2 + 12 x − 5 x − 20 .
Factor by grouping: 3 x ( x + 4 ) − 5 ( x + 4 ) .
Factor out the common factor: ( 3 x − 5 ) ( x + 4 ) . The final answer is ( 3 x − 5 ) ( x + 4 ) .
Explanation
Problem Analysis We are given the quadratic expression 3 x 2 + 7 x − 20 and we want to factor it. We will use the factoring by grouping method.
Finding the Right Numbers First, we need to find two numbers whose product is equal to the product of the leading coefficient and the constant term, which is 3 × ( − 20 ) = − 60 , and whose sum is equal to the middle coefficient, which is 7 .
Identifying the Numbers The two numbers that satisfy these conditions are 12 and − 5 , because 12 × ( − 5 ) = − 60 and 12 + ( − 5 ) = 7 .
Rewriting the Middle Term Now, we rewrite the middle term using these two numbers: 3 x 2 + 12 x − 5 x − 20 .
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 3 x 2 + 12 x ) + ( − 5 x − 20 ) . From the first group, we can factor out 3 x , and from the second group, we can factor out − 5 : 3 x ( x + 4 ) − 5 ( x + 4 ) .
Final Factorization Finally, we factor out the common factor ( x + 4 ) from both terms: ( 3 x − 5 ) ( x + 4 ) . Therefore, 3 x 2 + 7 x − 20 = ( 3 x − 5 ) ( x + 4 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, suppose you are designing a rectangular garden and you know the area can be represented by the expression 3 x 2 + 7 x − 20 . By factoring this expression into ( 3 x − 5 ) ( x + 4 ) , you can determine the possible dimensions of the garden in terms of x . This allows you to plan the layout and choose appropriate values for x based on the available space and desired proportions.
