Divide: $$\frac { - 8 + 9 x - x ^ { 2 } } { x ^ { 2 } - 10 x + 16 } \div \frac { x ^ { 2 } - 4 x + 3 } { x ^ { 2 } - 3 x + 2 }$$
A) $- ( x - 1 ) ( x - 3 )$
B) $\frac { x + 1 } { x + 3 }$
C) $\frac { x - 1 } { x - 3 }$
D) $- \frac { x - 1 } { x - 3 }$
E) $- \frac { ( x - 1 ) ( x - 3 ) } { ( x - 2 ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

To solve the given division of fractions, we first factorize all the polynomials and then simplify. The numerator of the first fraction: $-x^2 + 9x - 8$ can be factored as $-(x - 1)(x - 8)$.The denominator of the first fraction: $x^2 - 10x + 16$ can be factored as $(x - 2)(x - 8)$.The numerator of the second fraction: $x^2 - 4x + 3$ can be factored as $(x - 1)(x - 3)$.The denominator of the second fraction: $x^2 - 3x + 2$ can be factored as $(x - 1)(x - 2)$.So, the division becomes: $$\frac{-(x - 1)(x - 8)}{(x - 2)(x - 8)} \div \frac{(x - 1)(x - 3)}{(x - 1)(x - 2)}$$Dividing fractions is the same as multiplying by the reciprocal, so we get:$$\frac{-(x - 1)(x - 8)}{(x - 2)(x - 8)} \times \frac{(x - 1)(x - 2)}{(x - 1)(x - 3)}$$Simplifying by canceling out common factors, we are left with:$$-\frac{(x - 1)}{(x - 3)}$$Therefore, the correct answer is D) $- \frac { x - 1 } { x - 3 }$.

Divide −8+9x−x2x2−10x+16÷x2−4x+3x2−3x+2\frac { - 8 + 9 x - x ^ { 2 } } { x ^ { 2 } - 10 x + 16 } \div \frac { x ^ { 2 } - 4 x + 3 } { x ^ { 2 } - 3 x + 2 }x2−10x+16−8+9x−x2​÷x2−3x+2x2−4x+3​

Divide $\frac { - 8 + 9 x - x ^ { 2 } } { x ^ { 2 } - 10 x + 16 } \div \frac { x ^ { 2 } - 4 x + 3 } { x ^ { 2 } - 3 x + 2 }$