Divide the following: $\frac{7 z^4-3 z^2-z}{z^2}$
Answer (1)
Break the fraction into separate terms: z 2 7 z 4 − z 2 3 z 2 − z 2 z .
Simplify each term by subtracting the exponents.
Combine the simplified terms.
The final result is: 7 z 2 − 3 − z 1 .
Explanation
Understanding the Problem We are asked to divide the polynomial 7 z 4 − 3 z 2 − z by z 2 . The denominator is a monomial, so we can break up the fraction into separate terms.
Breaking up the Fraction We break up the fraction into separate terms: z 2 7 z 4 − 3 z 2 − z = z 2 7 z 4 − z 2 3 z 2 − z 2 z
Simplifying Each Term Now we simplify each term by dividing the numerator by the denominator. Recall that when dividing terms with the same base, we subtract the exponents: x b x a = x a − b .
For the first term: z 2 7 z 4 = 7 z 4 − 2 = 7 z 2 For the second term: z 2 3 z 2 = 3 z 2 − 2 = 3 z 0 = 3 For the third term: z 2 z = z 1 − 2 = z − 1 = z 1
Combining the Terms Finally, we combine the simplified terms: 7 z 2 − 3 − z 1
Final Answer The result of dividing z 2 7 z 4 − 3 z 2 − z is 7 z 2 − 3 − z 1 .
Examples
Understanding polynomial division is crucial in many areas, such as simplifying complex expressions in physics or engineering. For example, when analyzing the motion of an object under variable acceleration, you might encounter expressions that need simplification through division. Imagine calculating the trajectory of a rocket where the fuel consumption affects the acceleration; polynomial division helps in modeling and predicting the rocket's path accurately. This skill is also fundamental in computer graphics for rendering and manipulating 3D models efficiently.
