Use long division to find the quotient below.
[tex]$\left(16 x^3+40 x^2+72\right) \div(2 x+6)$[/tex]
A. [tex]$8 x^2+5 x+12$[/tex]
B. [tex]$8 x^2+3 x+12$[/tex]
C. [tex]$8 x^2-6 x+12$[/tex]
D. [tex]$8 x^2-4 x+12$[/tex]
Answer (2)
Divide 16 x 3 by 2 x to get 8 x 2 , then multiply ( 2 x + 6 ) by 8 x 2 and subtract from the dividend.
Divide − 8 x 2 by 2 x to get − 4 x , then multiply ( 2 x + 6 ) by − 4 x and subtract from the dividend.
Divide 24 x by 2 x to get 12 , then multiply ( 2 x + 6 ) by 12 and subtract from the dividend.
The quotient is 8 x 2 − 4 x + 12 .
Explanation
Understanding the Problem We are given the division problem ( 16 x 3 + 40 x 2 + 72 ) ÷ ( 2 x + 6 ) . We need to find the quotient using long division.
First Step of Long Division First, we set up the long division problem:
2x+6 | 16x^3 + 40x^2 + 0x + 72
We divide the leading term of the dividend ( 16 x 3 ) by the leading term of the divisor ( 2 x ): 2 x 16 x 3 = 8 x 2 . This is the first term of the quotient.
Subtracting the First Term Multiply the divisor ( 2 x + 6 ) by 8 x 2 : 8 x 2 ( 2 x + 6 ) = 16 x 3 + 48 x 2 . Subtract this from the dividend:
8x^2 ______
2x+6 | 16x^3 + 40x^2 + 0x + 72 -(16x^3 + 48x^2) ------------------ -8x^2 + 0x
Finding the Second Term Now, divide the leading term of the new dividend ( − 8 x 2 ) by the leading term of the divisor ( 2 x ): 2 x − 8 x 2 = − 4 x . This is the next term of the quotient.
Subtracting the Second Term Multiply the divisor ( 2 x + 6 ) by − 4 x : − 4 x ( 2 x + 6 ) = − 8 x 2 − 24 x . Subtract this from the new dividend:
8x^2 - 4x ____
2x+6 | 16x^3 + 40x^2 + 0x + 72 -(16x^3 + 48x^2) ------------------ -8x^2 + 0x -(-8x^2 - 24x) ------------------ 24x + 72
Finding the Third Term Now, divide the leading term of the new dividend ( 24 x ) by the leading term of the divisor ( 2 x ): 2 x 24 x = 12 . This is the next term of the quotient.
Subtracting the Third Term Multiply the divisor ( 2 x + 6 ) by 12 : 12 ( 2 x + 6 ) = 24 x + 72 . Subtract this from the new dividend:
8x^2 - 4x + 12
2x+6 | 16x^3 + 40x^2 + 0x + 72 -(16x^3 + 48x^2) ------------------ -8x^2 + 0x -(-8x^2 - 24x) ------------------ 24x + 72 -(24x + 72) ------------------ 0
Final Answer The remainder is 0. Therefore, the quotient is 8 x 2 − 4 x + 12 .
Conclusion The quotient of the division ( 16 x 3 + 40 x 2 + 72 ) ÷ ( 2 x + 6 ) is 8 x 2 − 4 x + 12 .
Examples
Polynomial long division isn't just a classroom exercise; it's used in engineering to design systems, in computer graphics to create curves, and in economics to model growth. For example, if you know the total cost function and the number of units produced, polynomial division can help you determine the average cost per unit. This technique provides a way to break down complex functions into simpler, more manageable parts, which is crucial for analysis and problem-solving in various fields.
The quotient of the division ( 16 x 3 + 40 x 2 + 72 ) ÷ ( 2 x + 6 ) is 8 x 2 − 4 x + 12 , which corresponds to option D. The long division process involves successive division, multiplication, and subtraction steps. Ultimately, the remainder is zero, confirming that the division was exact.
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