Divide the polynomials 2x3−x2−13x+22
by x2+x−5
to determine the quotient and remainder.(1 point)
Responses
The quotient is 2x−3
. The remainder is −20x+7
.
The quotient is 2 x minus 3. The remainder is negative 20 x plus 7.
The quotient is 2x−3
. The remainder is 7.
The quotient is 2 x minus 3. The remainder is 7.
The quotient is 2x+1
. The remainder is −4x+27
.
The quotient is 2 x plus 1. The remainder is negative 4 x plus 27.
The quotient is .2x+1
The remainder is 7.
The correct response is:
The quotient is 2x-3. The remainder is 7.
The correct answer is:
The quotient is 2x - 3. The remainder is 7.
To divide the polynomial 2x^3−x^2−13x+22 by x^2+x−5 and determine the quotient and remainder, you can use long division. Here are the steps to follow:
1. Arrange the polynomials in descending powers of x:
Divide 2x^3−x^2−13x+22 by x^2+x−5.
2. Divide the highest power of x in the dividend (2x^3) by the highest power of x in the divisor (x^2). The resulting term is the first term of the quotient:
Quotient term = 2x^(3-2) = 2x
3. Multiply the entire divisor (x^2+x−5) by the quotient term (2x), and subtract it from the original dividend. This step cancels out the x^3 term and reduces the degree of the dividend:
Remainder after subtracting: 2x^3−2x^2
4. Bring down the next term from the original dividend (-13x):
New dividend: 2x^3−2x^2−13x
5. Divide the highest power of x in the new dividend (2x^3) by the highest power of x in the divisor (x^2). The resulting term is the next term of the quotient:
Quotient term = 2x^(3-2) = 2x
6. Multiply the entire divisor (x^2+x−5) by the new quotient term (2x), and subtract it from the new dividend:
Remainder after subtracting: 2x^3−2x^2−13x−2x(x^2+x−5) = 2x^3−2x^2−13x−2x^3−2x^2+10x = -15x−2x^2+10x
7. Bring down the next term from the new dividend (22):
New dividend: -15x−2x^2+10x+22
8. Divide the highest power of x in the new dividend (-15x) by the highest power of x in the divisor (x^2). The resulting term is the next term of the quotient:
Quotient term = -15x^(1-2) = -15x^(-1) = -15/x
9. Multiply the entire divisor (x^2+x−5) by the new quotient term (-15/x), and subtract it from the new dividend:
Remainder after subtracting: -15x−2x^2+10x+22−(-15/x)(x^2+x−5) = -15x−2x^2+10x+22−(-15x−15−75/x) = -15x−2x^2+10x+22+15x+15+75/x = -2x^2+10x+37+75/x
10. There are no more terms left in the new dividend, so the division is complete. The final quotient is 2x−15/x, and the remainder is -2x^2+10x+37+75/x.
Therefore, the correct response is: "The quotient is 2x−15/x. The remainder is -2x^2+10x+37+75/x."