The quotient of a polynomial can be calculated by long division or by factoring the polynomial.
The quotient of [tex]\mathbf{\frac{2x^3-5x^2-3x+7}{x-2}}[/tex] is [tex]\mathbf{(a)\ 2x^2-x - 5}[/tex]
The expression is given as:
[tex]\mathbf{\frac{2x^3-5x^2-3x+7}{x-2}}[/tex]
First, we determine the remainder of the division
Let:
[tex]\mathbf{f(x) = 2x^3-5x^2-3x+7}[/tex]
Equate the divisor to 0
[tex]\mathbf{x - 2 = 0}[/tex]
Solve for x
[tex]\mathbf{x = 2}[/tex]
Substitute [tex]\mathbf{x = 2}[/tex] in f(x)
[tex]\mathbf{f(2) = 2(2)^3-5(2)^2-3(2)+7}[/tex]
[tex]\mathbf{f(2) = -3}[/tex]
So the remainder of [tex]\mathbf{\frac{2x^3-5x^2-3x+7}{x-2}}[/tex] is -3.
Subtract the remainder from f(x)
[tex]\mathbf{Quotient = \frac{2x^3-5x^2-3x+7+ 3}{x-2}}[/tex]
[tex]\mathbf{Quotient = \frac{2x^3-5x^2-3x+10}{x-2}}[/tex]
Factor the numerator
[tex]\mathbf{Quotient = \frac{(2x^2-x - 5)(x - 2)}{x-2}}[/tex]
Cancel out the common factors
[tex]\mathbf{Quotient = 2x^2-x - 5}[/tex]
Hence, the quotient is [tex]\mathbf{(a)\ 2x^2-x - 5}[/tex]
Read more about quotients at:
https://brainly.com/question/6813716
