(a) verify the given factor(s) of the function f, (b) find...

(a) verify the given factor(s) of the function $f,$ (b) find the remaining factors of $f,(c)$ use your results to write the complete factorization of $f,$ and (d) list all real zeros of $f$. Confirm your results by using a graphing utility to graph the function. Function $$\begin{aligned} f(x)=x^{4} &-4 x^{3}-15 x^{2} \\ &+58 x-40 \end{aligned}$$ Factor $(s)$ $$(x-5),(x+4)$$

(a) verify the given factor(s) of the function $f,$ (b) find the remaining factors of $f,(c)$ use your results to write the complete factorization of $f,$ and (d) list all real zeros of $f$. Confirm your results by using a graphing utility to graph the function.
Function $$\begin{aligned} f(x)=x^{4} &-4 x^{3}-15 x^{2} \\ &+58 x-40 \end{aligned}$$ Factor $(s)$ $$(x-5),(x+4)$$

Key Concepts

Graphing Utility Verification

Using a graphing utility to verify the factorization involves plotting the polynomial function to visually confirm that the identified zeros are correct and that the graph intersects the x-axis at these points. This serves as an important check for the correctness of the algebraic factorization.

Real Zeros of a Polynomial

The real zeros of a polynomial are the values of x for which the polynomial equals zero. Identifying these zeros involves solving the factored equations and observing their multiplicity, and they provide critical information about the graph of the polynomial.

Complete Factorization

Complete factorization involves expressing a polynomial as a product of its irreducible factors, often combining various methods such as verifying factors, division, and further factorization of lower degree polynomials.

Polynomial Division

Polynomial division, including synthetic division or long division, is used to divide a polynomial by a known factor. This process simplifies the polynomial by reducing its degree and helps in finding the remaining factors.

Factor Theorem

The Factor Theorem states that if a polynomial f(x) has f(a)=0, then (x?a) is a factor of f(x). This concept is used to verify whether a given binomial factor divides the polynomial without a remainder.

Transcript

00:02 Given the function f of x equals x to the 4 minus 4x cubed minus 15x squared plus 58x minus 40 and the factors of x minus 5 and x plus 4, we're going to first verify that these are in fact factors of this function and we're going to do so using synthetic division.

00:21 So first we're going to say x minus 5 equals 0 and x plus 4 equals 0.

00:27 That's what it means to be a factor of a function, that part of the factor equals 0.

00:32 I'm going to add five to both sides of this one, and we get x equals five.

00:37 I'm going to subtract four from both sides of this one, and we get x equals negative four.

00:43 I'm going to set up my synthetic division for the first one.

00:47 Inside of this box, we just put our coefficients of each term, 1x to the 4 minus 4x cubed, minus 15x squared, plus 58x minus 40.

01:00 Outside of the box here, i'm going to start with the five.

01:04 This first number here just always drops straight down.

01:07 So i'm going to put a one there.

01:09 Anything in this bottom row outside of the box, we're going to multiply by this number outside of the box.

01:15 Five times one is five.

01:16 And we put that there.

01:18 And we get negative four plus five, which is just one.

01:27 Then we do five times one again.

01:29 And we get five.

01:31 Negative 15 plus five is negative 10.

01:35 Five times negative 10 is negative 50.

01:37 58 minus 10 is 8 and 5 times 8 is 40 and negative 40 plus 40 is 0.

01:47 Since we ended with a zero here, that means this is in fact a factor of this function.

01:55 We're going to do the same thing.

01:58 Use synthetic division to verify that negative 4 that we got...

Please give Ace some feedback