What is the completely factored form of the expression [tex]$16 x^2+8 x+32$[/tex]?
A. [tex]$4(4 x^2+2 x+8)$[/tex]
B. [tex]$4(12 x^2+4 x+28)$[/tex]
C. [tex]$8(2 x^2+x+4)$[/tex]
D. [tex]$8 x(8 x^2+x+24)$[/tex]
Answer (1)
Find the greatest common factor (GCF) of the coefficients: The GCF of 16, 8, and 32 is 8.
Factor out the GCF: 16 x 2 + 8 x + 32 = 8 ( 2 x 2 + x + 4 ) .
Check the discriminant of the quadratic 2 x 2 + x + 4 : b 2 − 4 a c = 1 2 − 4 ( 2 ) ( 4 ) = − 31 .
Since the discriminant is negative, the quadratic cannot be factored further. The completely factored form is 8 ( 2 x 2 + x + 4 ) .
Explanation
Problem Analysis We are given the expression 16 x 2 + 8 x + 32 and asked to factor it completely.
Finding the GCF First, we find the greatest common factor (GCF) of the coefficients 16, 8, and 32. The GCF is 8.
Factoring out the GCF We factor out the GCF from the expression: 16 x 2 + 8 x + 32 = 8 ( 2 x 2 + x + 4 ) .
Checking the Discriminant Now, we check if the quadratic expression 2 x 2 + x + 4 can be factored further. To do this, we examine its discriminant, which is given by b 2 − 4 a c , where a = 2 , b = 1 , and c = 4 . The discriminant is 1 2 − 4 ( 2 ) ( 4 ) = 1 − 32 = − 31 . Since the discriminant is negative, the quadratic expression 2 x 2 + x + 4 cannot be factored further using real numbers.
Final Factored Form Therefore, the completely factored form of the given expression is 8 ( 2 x 2 + x + 4 ) .
Examples
Factoring quadratic expressions is useful in many real-world applications. For example, engineers use factoring to design structures and analyze stress. Imagine you are designing a bridge and need to ensure it can withstand certain loads. By expressing the load distribution as a quadratic equation and factoring it, you can identify critical points and reinforce those areas to prevent structural failure. This ensures the bridge is safe and durable.
