What is the factorization of the trinomial below?
[tex]x^3-12 x^2+35 x[/tex]
A. [tex](x-7)(x+5)[/tex]
B. [tex](x^2-7)(x+5)[/tex]
C. [tex]x(x-7)(x+5)[/tex]
D. [tex]x(x-7)(x-5)[/tex]
Answer (1)
Factor out the common factor x : x 3 − 12 x 2 + 35 x = x ( x 2 − 12 x + 35 ) .
Factor the quadratic expression x 2 − 12 x + 35 by finding two numbers that multiply to 35 and add to -12, which are -5 and -7.
Write the quadratic expression as ( x − 5 ) ( x − 7 ) .
The complete factorization is x ( x − 5 ) ( x − 7 ) , so the answer is x ( x − 7 ) ( x − 5 ) .
Explanation
Understanding the Problem We are given the trinomial x 3 − 12 x 2 + 35 x and asked to find its factorization.
Factoring out the Common Factor First, we observe that each term in the trinomial has a factor of x . We can factor out x from the entire expression: x 3 − 12 x 2 + 35 x = x ( x 2 − 12 x + 35 ) Now we need to factor the quadratic expression x 2 − 12 x + 35 .
Factoring the Quadratic Expression We are looking for two numbers that multiply to 35 and add up to -12. The factors of 35 are 1 and 35, or 5 and 7. Since the middle term is -12, we need two negative numbers. The numbers -5 and -7 satisfy the conditions since ( − 5 ) × ( − 7 ) = 35 and ( − 5 ) + ( − 7 ) = − 12 .
Complete Factorization Therefore, we can write the quadratic expression as ( x − 5 ) ( x − 7 ) . So the complete factorization of the trinomial is: x ( x − 5 ) ( x − 7 ) This matches option D.
Examples
Factoring trinomials is a fundamental skill in algebra and has practical applications in various fields. For example, engineers use factoring to simplify complex equations when designing structures or analyzing systems. Architects use factoring to optimize space and materials in building designs. Even in finance, factoring can help simplify calculations when analyzing investments or predicting market trends. Understanding factoring empowers you to solve real-world problems efficiently and accurately.
