Factor this polynomial completely.
[tex]15 x^2-11 x-12[/tex]
A. [tex](15 x-3)(x+4)[/tex]
B. [tex](5 x+3)(3 x-4)[/tex]
C. [tex](15 x-4)(x+3)[/tex]
D. [tex](3 x+3)(5 x-4)[/tex]
Answer (2)
• Find two numbers whose product is 15 × − 12 = − 180 and whose sum is − 11 , which are − 20 and 9 .
• Rewrite the middle term: 15 x 2 − 11 x − 12 = 15 x 2 − 20 x + 9 x − 12 .
• Factor by grouping: 15 x 2 − 20 x + 9 x − 12 = 5 x ( 3 x − 4 ) + 3 ( 3 x − 4 ) .
• Factor out the common factor: ( 5 x + 3 ) ( 3 x − 4 ) . The factored form is ( 5 x + 3 ) ( 3 x − 4 ) .
Explanation
Problem Analysis We are given the quadratic polynomial 15 x 2 − 11 x − 12 and asked to factor it completely.
Finding the Right Numbers To factor the quadratic polynomial, we need to find 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 × − 12 = − 180 and whose sum is − 11 .
Identifying the Numbers The two numbers that satisfy these conditions 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: 15 x 2 − 11 x − 12 = 15 x 2 − 20 x + 9 x − 12 .
Factoring by Grouping Next, we factor by grouping: 15 x 2 − 20 x + 9 x − 12 = 5 x ( 3 x − 4 ) + 3 ( 3 x − 4 ) .
Final Factorization Finally, we factor out the common factor ( 3 x − 4 ) : 5 x ( 3 x − 4 ) + 3 ( 3 x − 4 ) = ( 5 x + 3 ) ( 3 x − 4 ) . Therefore, the factored form of the polynomial is ( 5 x + 3 ) ( 3 x − 4 ) .
Examples
Factoring quadratic polynomials is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Imagine you are designing a rectangular garden with an area represented by the quadratic expression 15 x 2 − 11 x − 12 . By factoring this expression into ( 5 x + 3 ) ( 3 x − 4 ) , you can determine the possible dimensions of the garden in terms of x . This allows you to optimize the garden's layout based on the available space and desired area.
The polynomial 15 x 2 − 11 x − 12 factors completely to ( 5 x + 3 ) ( 3 x − 4 ) . This is achieved by finding two numbers that multiply to − 180 and add to − 11 , breaking the middle term, grouping, and factoring. The answer is B : ( 5 x + 3 ) ( 3 x − 4 ) .
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