(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. $f(x)=10 x^{3}-11 x^{2}-72 x+45 \quad(2 x+5),(5 x-3)$

(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.
$f(x)=10 x^{3}-11 x^{2}-72 x+45 \quad(2 x+5),(5 x-3)$

Key Concepts

Factor Theorem

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. This theorem is fundamental in polynomial factorization, as it allows one to verify whether a given linear expression is indeed a factor of the polynomial by simply evaluating the polynomial at the corresponding value.

Polynomial Division

Polynomial division, including long division and synthetic division, is a method used to divide a polynomial by a binomial or another polynomial. This process is essential when breaking down a higher degree polynomial into its factors, as it helps in finding the remaining factors once some factors are known.

Remainder Theorem

The Remainder Theorem is closely linked to the Factor Theorem and states that when a polynomial f(x) is divided by (x - c), the remainder is f(c). This theorem provides an efficient way to check for factors and to perform polynomial corrections during the factorization process.

Zero Finding and Complete Factorization

Finding the zeros (or roots) of a polynomial involves determining the values of x for which f(x) = 0, which correspond directly to the factors of the polynomial. Complete factorization decomposes the polynomial into products of linear factors (or irreducible polynomials), making it easier to understand the behavior of the polynomial and solve polynomial equations.

Graphical Analysis of Polynomials

Graphical analysis involves using graphing tools to plot the polynomial function, which helps in verifying the identified zeros and factors visually. The intersections of the graph with the x-axis indicate the real zeros, thus providing a confirmation of the factorization and the results obtained from algebraic methods.

Transcript

00:01 Given this function f and two factors from it, notice that this is a degree three polynomial, which means it's going to have three root, possibly three root three factors.

00:11 We're given two of them already.

00:13 Our first order of business is to verify that these are factors.

00:17 And we can do this by recognizing the k value that would go along with them, which is the solutions would be negative five halves and three fifths.

00:27 Imagine them being the zeros.

00:33 And so it's true that if we, well, let's find out.

00:38 What happens when we plug negative five halves into this function? when we get f of negative five halves equals 10 times negative five halves cubed minus 11 times negative five halves squared minus 72 times negative five halves was 45.

00:55 And this equals zero.

01:00 And if we try the three -fifths, it would be plugging three -fifths in for x, and that as well, check your calculator, should equal zero.

01:23 So they're both factors.

01:26 We can find the third one by dividing these known factors out, and we should get remainders of zero.

01:33 So first, let's use the 2x plus 5.

01:36 Because this isn't of the x -minus k nature, we don't want to do synthetic division.

01:39 We're going to do long division.

01:40 I'm going to divide that from our polynomial 10x cubed minus 11 x squared minus 72x plus 45.

01:53 Here we go.

01:54 Divide 10x cubed by 2x.

01:56 We get 5x squared.

02:01 Distribute, we get 10x cube plus 25x squared.

02:07 Then we subtract 10x cubes cancel.

02:09 Negative 11 minus 25 is negative 36x squared.

02:13 Drop down divide negative 36 divided by negative 2 is negative 18 x distribute we get negative 36 x squared and 18 times 5 is 90 so we'd get minus 90 x subtract negative 72 minus negative 90 is like negative 72 plus 90 and we get 18x drop down the 40 do one last time, 18x divided by 2x is 9.

02:53 Distribute, we get 18 x plus 45.

02:57 And sure enough, when we subtract, it all cancels...

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