Create a unique example of dividing a polynomial by a monomial and provide the simplified form. Explain, in complete sentences, the two ways used to simplify this expression and how you would check your quotient for accuracy.
please explain :/ i only remember one way to solve this
Sure! Let's say we have the polynomial 3x^3 + 6x^2 + 9x divided by the monomial 3x.
To simplify this expression, we can use two methods: the distributive property and canceling common factors.
Method 1: Using the Distributive Property
To divide a polynomial by a monomial using the distributive property, we divide each term of the polynomial by the monomial. In our example:
Term 1: (3x^3) / (3x) = x^2
Term 2: (6x^2) / (3x) = 2x
Term 3: (9x) / (3x) = 3
Therefore, our simplified quotient is x^2 + 2x + 3.
Method 2: Canceling Common Factors
Alternatively, we can simplify the expression by canceling out the common factor between the polynomial terms and the monomial. In this case, the common factor is 3x. By factoring it out, we have:
3x^3 + 6x^2 + 9x = 3x(x^2 + 2x + 3)
Therefore, our simplified quotient is also x^2 + 2x + 3.
To check the accuracy of our quotient, we can multiply the simplified quotient by the monomial and see if we obtain the original polynomial. In this case, we can multiply (x^2 + 2x + 3) by 3x and check if we get the original polynomial 3x^3 + 6x^2 + 9x. If they match, our quotient is accurate.