(a) verify the given factors of f(x) (b) find the remaining...

(a) verify the given factors of $f(x)$ (b) find the remaining factor(s) of $f(x)$ (c) use your results to write the complete factorization of $f(x),$ (d) list all real zeros of $f,$ and (e) confirm your results by using a graphing utility to graph the function. $$\begin{array}{ll}\text { Function } & \text { Factors } \\f(x)=2 x^{3}-x^{2}-10 x+5 & (2 x-1),(x+\sqrt{5})\end{array}$$

(a) verify the given factors of $f(x)$
(b) find the remaining factor(s) of $f(x)$
(c) use your results to write the complete factorization of $f(x),$
(d) list all real zeros of $f,$ and
(e) confirm your results by using a graphing utility to graph the function.
$$\begin{array}{ll}\text { Function } & \text { Factors } \\f(x)=2 x^{3}-x^{2}-10 x+5 & (2 x-1),(x+\sqrt{5})\end{array}$$

Key Concepts

Graphical Analysis

Graphical analysis entails plotting the polynomial function to visually inspect the locations of its zeros and its overall behavior. This approach provides a practical method for verifying the results obtained algebraically, ensuring consistency between the factorization and the function's graphical representation.

Synthetic Division

Synthetic division is an efficient alternative to the traditional long division method when dividing polynomials by linear factors. It simplifies calculations, making it quicker to find the quotient and remainder, which is particularly useful in the factorization process and in confirming candidate factors.

Zeros of a Polynomial

The zeros of a polynomial are the solutions to the equation f(x) = 0; they represent the values of x where the polynomial crosses or touches the x-axis. Identifying these zeros is crucial for expressing the polynomial in its fully factored form and understanding its behavior.

Factor Theorem

The factor theorem establishes that a polynomial f(x) has a factor of the form (x - c) if and only if f(c) equals zero. This theorem is central to polynomial factorization because it provides a method to verify whether a given candidate is indeed a factor based on simple substitution.

Polynomial Division

Polynomial division is a systematic method for dividing a polynomial by another polynomial, typically of lower degree, to determine the quotient and remainder. This process, which can be executed via the long division algorithm, is essential for breaking down higher degree polynomials into simpler, more manageable factors.

Transcript

00:01 One first we need to verify the given factors to do that we need to do long division so my first divisor is the first factor 2x minus 1 and i'm going to divide the entire polynomial which is 2x cubed minus x squared minus 10x plus 5 all right so you have to think how many times is 2x go into 2x cubed.

00:25 All right.

00:27 I'm sorry, you need to think how many, how many more xs do you need to make 2x go into 2x cubed, and you need two more x.

00:34 Distribute that to both.

00:36 So you got 2x cubed minus x squared.

00:40 All right, we subtract the whole quantity, which then changes our signs, and everything actually cancels.

00:45 So i'm going to drop down my next two terms.

00:50 All right, so now we need to think, what do i need to multiply 2x by to make it? negative 10x.

00:57 That would be a negative 5.

01:00 Distribute that to both.

01:01 So i'll get a negative 10x plus 5, subtract with quantities, which changes my signs, and i end up with zero.

01:09 Since my remainder is zero, that means it goes in evenly, which therefore does confirm that it is a factor.

01:15 All right.

01:16 Next, i'm going to continue dividing using my other factor, which is x plus root 5, into my answer, which was x squared, minus 5, but i do need to put in the placeholder of the 0x.

01:37 All right, so once again, think about what do we need to multiply x by to make it x squared? so i just need an x, distribute that to both.

01:44 So i get x squared plus x root 5 subtract, so i'm going to change my signs...

Please give Ace some feedback