Use long division to find the quotient below.
$\left(4 x^2-7 x-2\right)-(x-2)$
A. $4 x+1$
B. $3 x+1$
C. $4 x-1$
D. $3 x-1$
Answer (1)
Perform polynomial long division to divide ( 4 x 2 − 7 x − 2 ) by ( x − 2 ) .
The quotient is found to be 4 x + 1 .
Divide 4 x 2 by x to get 4 x , then multiply ( x − 2 ) by 4 x and subtract from the dividend.
Divide x by x to get 1 , then multiply ( x − 2 ) by 1 and subtract from the remaining dividend.
The final quotient is 4 x + 1 .
Explanation
Understanding the Problem We are given the division problem ( 4 x 2 − 7 x − 2 ) ( x − 2 ) and asked to find the quotient using long division.
Setting up Long Division To perform polynomial long division, we set up the problem as follows:
x - 2 | 4x^2 - 7x - 2
We divide the first term of the dividend, 4 x 2 , by the first term of the divisor, x . This gives us 4 x , which is the first term of the quotient.
First Term of Quotient We multiply the divisor, x − 2 , by 4 x to get 4 x ( x − 2 ) = 4 x 2 − 8 x . We subtract this from the dividend:
4x
x - 2 | 4x^2 - 7x - 2
-(4x^2 - 8x)
-----------
x - 2
Bringing Down the Next Term Now we bring down the next term, which is − 2 . We divide the first term of the new dividend, x , by the first term of the divisor, x . This gives us 1 , which is the second term of the quotient.
Second Term of Quotient We multiply the divisor, x − 2 , by 1 to get 1 ( x − 2 ) = x − 2 . We subtract this from the new dividend:
4x + 1
x - 2 | 4x^2 - 7x - 2
-(4x^2 - 8x)
-----------
x - 2
-(x - 2)
---------
0
Final Result The remainder is 0 , so the quotient is 4 x + 1 .
Examples
Polynomial long division is a method used to divide polynomials, similar to long division with numbers. It's useful in various applications, such as simplifying rational expressions, finding roots of polynomials, and solving algebraic equations. For example, if you know one factor of a polynomial, you can use long division to find the other factors. This technique is also used in calculus when integrating rational functions.
