What is the quotient in simplest form? State the restrictions on the variable. (z^2-4)/(z-3) divided by (z+2)/(z^2+z-12) I got z^2+2z-8, but I am not sure of the restrictions. Maybe 3 and -2.
The Correct Answer and Explanation is:
Let’s break this down step by step.
You are given the expression:z2−4z−3÷z+2z2+z−12\frac{{z^2 – 4}}{{z – 3}} \div \frac{{z + 2}}{{z^2 + z – 12}}z−3z2−4÷z2+z−12z+2
To divide fractions, we multiply by the reciprocal of the second fraction. So, this becomes:z2−4z−3×z2+z−12z+2\frac{{z^2 – 4}}{{z – 3}} \times \frac{{z^2 + z – 12}}{{z + 2}}z−3z2−4×z+2z2+z−12
Step 1: Factor the terms
- z2−4z^2 – 4z2−4 is a difference of squares, so it factors as:
z2−4=(z−2)(z+2)z^2 – 4 = (z – 2)(z + 2)z2−4=(z−2)(z+2)
- z2+z−12z^2 + z – 12z2+z−12 can be factored by looking for two numbers that multiply to -12 and add up to 1. These numbers are 4 and -3, so we have:
z2+z−12=(z+4)(z−3)z^2 + z – 12 = (z + 4)(z – 3)z2+z−12=(z+4)(z−3)
Now, substitute these factored forms into the expression:(z−2)(z+2)z−3×(z+4)(z−3)z+2\frac{{(z – 2)(z + 2)}}{{z – 3}} \times \frac{{(z + 4)(z – 3)}}{{z + 2}}z−3(z−2)(z+2)×z+2(z+4)(z−3)
Step 2: Cancel out common factors
Notice that z+2z + 2z+2 appears in both the numerator and denominator, and z−3z – 3z−3 also appears in both the numerator and denominator. So, we can cancel these terms:=(z−2)(z+4)1= \frac{{(z – 2)(z + 4)}}{1}=1(z−2)(z+4)
Which simplifies to:=(z−2)(z+4)= (z – 2)(z + 4)=(z−2)(z+4)
Expanding this:=z2+4z−2z−8= z^2 + 4z – 2z – 8=z2+4z−2z−8=z2+2z−8= z^2 + 2z – 8=z2+2z−8
So, your result is correct: the quotient is z2+2z−8z^2 + 2z – 8z2+2z−8.
Step 3: Find the restrictions on the variable
To find the restrictions, we must look at the values of zzz that would make any denominator equal to zero. From the original expression, the denominators are z−3z – 3z−3 and z+2z + 2z+2, as well as the denominator z2+z−12z^2 + z – 12z2+z−12 that factored to (z−3)(z+4)(z – 3)(z + 4)(z−3)(z+4).
- z−3=0z – 3 = 0z−3=0 implies z=3z = 3z=3
- z+2=0z + 2 = 0z+2=0 implies z=−2z = -2z=−2
- z+4=0z + 4 = 0z+4=0 implies z=−4z = -4z=−4
Thus, the restrictions are z≠3z \neq 3z=3, z≠−2z \neq -2z=−2, and z≠−4z \neq -4z=−4, because these values would make the denominator zero and the expression undefined.
Final Answer:
- Quotient: z2+2z−8z^2 + 2z – 8z2+2z−8
- Restrictions: z≠3z \neq 3z=3, z≠−2z \neq -2z=−2, and z≠−4z \neq -4z=−4.
