Question

Show that the binomial is a factor of the polynomial. Then factor the function completely. $$ g(x)=x^3-x^2-20 x ; x+4 $$

Short answer

The binomial (x + 4) is a factor of the polynomial. The function can be completely factored as follows: \(g(x)= x^3 - x^2 - 20x = (x + 4)(x^2 - 5x - 5)\)

Step-by-step solution

Verify if Binomial is a Factor

First, we need to test if the binomial (x + 4) is a factor of the polynomial, g(x). This can be done by substituting x = -4 into g(x). Thus, we test g(-4), if this becomes zero then the binomial is a factor.\( g(-4) = (-4)^3 - (-4)^2 - 20*(-4) = -64 - 16 + 80 = 0 \). Since g(-4) equals zero, we can confirm that (x + 4) is indeed a factor of g(x).

Long Division

After confirming that (x+4) is a factor, we can find other factors. This can be achieved through long division by dividing the polynomial by the binomial. We divide \( x^3 - x^2 - 20x \) by \( x + 4 \) and obtain the quotient \( x^2 - 5x - 5 \). This quotient is also a factor of the original polynomial.

Factorize the Quotient

Next step is to attempt to factorize the quotient further. In this case, the quadratic equation in step 2, \( x^2 - 5x - 5 \), cannot be factorized further as it doesn't have real roots.

Final Factorized form

Now, we express the original polynomial in its completely factored form. We get \(g(x)= x^3 - x^2 - 20x = (x + 4)(x^2 - 5x - 5)\). As the quadratic equation cannot be factorized further, this is the final factorized form of the polynomial.

Key concepts

Binomial Factors

Understanding how to identify binomial factors in a polynomial is an essential skill in algebra. A binomial is a two-term algebraic expression such as \(x + 4\). In assessing whether a binomial is a factor of a polynomial, one useful method is to apply the Factor Theorem. According to this principle, if substituting \(x = -a\) into the polynomial \(g(x)\) results in \(g(-a) = 0\), then \(x + a\) is a factor of \(g(x)\).

For instance, when we evaluate \(g(x) = x^3 - x^2 - 20x\) at \(x = -4\), and it yields zero, \(x + 4\) confirms itself as a factor. This fundamental insight paves the way to breaking down complex polynomials into more manageable pieces, ultimately simplifying the polynomial into its factored form. This process not only provides clarity in understanding the structure of the polynomial but also is a stepping stone to solving equations or graphing the polynomial function.

Long Division of Polynomials

Long division of polynomials is akin to the long division process you may recall from arithmetic, but with a twist for handling variables and degrees of polynomials. This process is crucial when decomposing a polynomial into its component factors, especially after determining that a binomial is a factor using the Factor Theorem.

Let's take through the steps. To divide \(x^3 - x^2 - 20x\) by \(x + 4\), we repeatedly subtract multiples of \(x + 4\) from \(x^3 - x^2 - 20x\) until we're left with a result that has a lower degree than \(x + 4\). The outcome of this division gives us the quotient which, in our case, is \(x^2 - 5x - 5\). If performed correctly, long division also confirms the binomial as a factor of the polynomial and facilitates the uncovering of more complex factorization structures.

Polynomial Roots

Polynomial roots are values of \(x\) that make the polynomial equation equal to zero. These roots hold significant implications for the graph and the behavior of the polynomial function. In our context, after successfully dividing the polynomial \(g(x)\) by \(x + 4\), we are left with \(x^2 - 5x - 5\). To find its roots, one would typically factorize the quadratic, but if it can't be factored using real numbers, as in this case, one may resort to the quadratic formula or other numerical methods.

In such instances where the roots cannot be represented by real numbers, they are real-world manifestations of the underlying mathematics indicating a lack of x-intercepts on the graph of the polynomial. Understanding the roots of polynomials, whether real or complex, is pivotal in solving equations and comprehending the intrinsic properties of polynomial functions.