In Exercises 1-6, divide f(x) by d(x), and write a summary statement in polynomial form and frac

step 1 Okay, for this question we have F of x divided by D of X. F of X is X cubed plus 4x squared plus

In Exercises 1-6, divide f(x) by d(x), and write a summary...

Key Concepts

Fractional Representation

Besides expressing the result in polynomial form, the division result can also be represented in fractional form. In this representation, the remainder is written as a fraction with the original divisor as the denominator, effectively expressing the original polynomial as a sum of the quotient and a proper fraction.

Quotient and Remainder

In polynomial division, the quotient is the polynomial that results from the division process, while the remainder is the polynomial leftover that has a degree less than the divisor. The division is often summarized by the formula: dividend = (divisor × quotient) + remainder, which is analogous to the division algorithm for integers.

Polynomial Division

Polynomial division is a process similar to arithmetic long division where one polynomial (the dividend) is divided by another polynomial (the divisor) to yield a quotient and possibly a remainder. This method is fundamental in algebra for simplifying expressions, solving polynomial equations, and performing partial fraction decomposition.

Long Division Algorithm

The long division algorithm for polynomials involves dividing the leading term of the dividend by the leading term of the divisor to determine each term of the quotient. This method is performed iteratively, subtracting the product of the divisor and the current quotient term from the dividend segment, and then repeating the process with the resulting polynomial.

Transcript

00:05 Okay, for this question we have f of x divided by d of x.

00:13 F of x is x cubed plus 4x squared plus 7x minus 9, and that's going to be divided by d of x, which is x plus 3.

00:32 So first thing we need to do is non -permissible values, meaning numbers that will make the denominator equal to zero and therefore make the function.

00:43 Undefined.

00:44 So if x plus 3 can't equal 0, then that means that x cannot equal negative 3, because if i sub in negative 3 plus 3, that will of course make 0, making the denominator equal to 0.

00:58 So we'll just keep that over there for now.

01:01 So now what i need to do is i need to divide these two things.

01:04 Now i can't factor this one very easily.

01:07 What i actually need to do is long division or synthetic division.

01:11 I'm going to do long division.

01:14 So we set it up just like back in grade school when you were doing long division of numbers, only this time we are doing long division of polynomials.

01:28 So i set it up just the same.

01:34 And now what i'm going to do is i'm going to look at the first one.

01:36 So i have x.

01:39 And what i need to do is figure out what is x cubed divided by x.

01:47 So that of course would be x squared...

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