DISCOVER: Nested Form of a Polynomial Expand Q to prove that...

DISCOVER: Nested Form of a Polynomial Expand Q to prove that the polynomials $P$ and $Q$ are the same. $$P(x)=3 x^{4}-5 x^{3}+x^{2}-3 x+5$$ $$Q(x)=(((3 x-5) x+1) x-3) x+5$$ Try to evaluate $P(2)$ and $Q(2)$ in your head, using the forms given. Which is easier? Now write the polynomial $R(x)=x^{5}-2 x^{4}+3 x^{3}-2 x^{2}+3 x+4$ in "nested" form, like the polynomial $Q .$ Use the nested form to find $R(3)$ in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value of a polynomial using synthetic division?

DISCOVER: Nested Form of a Polynomial Expand Q to
prove that the polynomials $P$ and $Q$ are the same.
$$P(x)=3 x^{4}-5 x^{3}+x^{2}-3 x+5$$
$$Q(x)=(((3 x-5) x+1) x-3) x+5$$
Try to evaluate $P(2)$ and $Q(2)$ in your head, using the
forms given. Which is easier? Now write the polynomial
$R(x)=x^{5}-2 x^{4}+3 x^{3}-2 x^{2}+3 x+4$ in "nested" form,
like the polynomial $Q .$ Use the nested form to find $R(3)$ in
your head. Do you see how calculating with the nested form follows
the same arithmetic steps as calculating the value of a polynomial using synthetic division?

Key Concepts

Polynomial Representation

This concept deals with expressing a polynomial in different forms, such as the standard expanded form, where each term is written separately with its corresponding power of the variable, and the nested form, where the expression is restructured to highlight the repeated multiplication. Understanding different representations can often simplify computations and proofs related to the polynomial.

Nested Form (Horner's Method)

The nested form restructures a polynomial into a format that allows for efficient evaluation. Also known as Horner's method, this approach rewrites the polynomial so that it becomes a series of multiplications and additions. It greatly simplifies calculations, especially when evaluating the polynomial at a certain point, by reducing the number of arithmetic operations required.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form (x - c). It uses the coefficients of the polynomial to perform the division with less effort compared to long division. The arithmetic steps involved are similar to those in the nested form evaluation, and it is particularly useful for evaluating polynomials and finding roots.

Polynomial Evaluation

This concept involves calculating the value of a polynomial for a given input. Using either the standard form or the nested form (Horner’s method), one can substitute a number into the polynomial and perform the necessary arithmetic operations. The nested form is particularly advantageous because it streamlines the process, making mental calculations more manageable.

Transcript

00:01 Okay, a few things going on in this problem.

00:02 First, we have to show that these two are equal.

00:05 And it's actually really not that hard to do because the only thing you have to distribute is the x.

00:11 So we'll distribute x to both the five and to the three x.

00:15 So this becomes three x squared, minus five x, and then plus one is multiplied by x, minus three is multiplied by x, and then plus five.

00:29 So let me match up the amount of parentheses, which is just, 2 now and then you take x and you just multiply by all these things x times 3x squared is 3x to the 3rd x times negative 5x squared x times 1 is xx2x still got a minus 3 3 3 3000 and that whole thing is going to be multiplied by x that's all that's left in our parentheses all right so it's multiply x times each of these individually x times 3x to the 3rd is 3x to the 4th x times negative 5 x squared is negative 5 x to the third x times x is x squared and x times a negative 3 is a negative 3x and we see that indeed they are exactly the same after distributing those x's to a few of the terms so now the next thing they ask this to do i'm going to go ahead and erase this so it's out of the way is in our head to plug in basically a two for this right here here and see what you get.

01:42 And so i'm going to go ahead and plug in that two and see what i get.

01:45 So i get a two times a three.

01:47 Two times three is six.

01:48 Six minus five is one.

01:51 One times two is two.

01:53 Two plus one is three.

01:55 Three times two is six.

01:57 Six minus three is three.

02:00 Three times two is six and six plus five is eleven.

02:05 So i was able to do that one step at a time pretty easily.

02:09 Step is they said, okay, now take this and put it in that nested form.

02:14 And so i'm going to be very careful as i do this because this is my first time kind of seeing this nested form.

02:20 It's very interesting, but what you should recognize is that there are no exponents anywhere.

02:26 So see that x of the fifth, it just becomes x minus and you keep the coefficients.

02:32 So i'll just put one parentheses here, but then what's distributed it every time is just an x.

02:40 And so you can see that plus three, i need to keep that coefficient, put a parenthesis and only multiply by an x here, and then put a minus two.

02:51 Again, parentheses only put an x and then put a plus three...

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