Question

Show that the binomial is a factor of the polynomial. Then factor the function completely. $$ f(x)=x^4-6 x^3-8 x+48 ; x-6 $$

Short answer

The binomial \(x - 6\) is a factor of the polynomial \(f(x) = x^4 - 6x^3 - 8x + 48\). After complete factorization, \(f(x)\) is expressed as \(f(x) = (x - 6)(x^3 + 4)\).

Step-by-step solution

Verify binomial as factor

By the Factor theorem, a polynomial \(f(x)\) has a factor \(x-c\) if and only if \(f(c)=0\). Substituting \(x = 6\) into the polynomial \(f(x)=x^4-6x^3-8x+48\), obtain \(f(6)\). If \(f(6) = 0\), \(x - 6\) is a factor of the polynomial.

Calculate \(f(6)\)

Substitute \(x = 6\) into \(f(x)\) to get \(f(6)= (6)^4 - 6(6)^3-8*(6) +48 = 36 - 216 - 48 + 48 = 0\). Since \(f(6) = 0\), \(x - 6\) is a factor of the polynomial.

Factorize the Polynomial

To factorize the Polynomial, Now, perform long division or synthetic division on the given function by dividing it by \(x-6\). The quotient obtained gives the remaining factors of the polynomial.

Perform Division

On dividing \(x^4-6x^3-8x+48\) by \(x-6\), we get the quotient \(x^3 + 4\). Hence, the function \(f(x)\) can be factored completely as \(f(x) = (x - 6)(x^3 + 4)\).

Key concepts

Factor Theorem

The Factor Theorem is a fundamental concept in algebra and helps us identify whether a binomial is a factor of a given polynomial. It states that a polynomial \(f(x)\) has a factor \(x-c\) if and only if \(f(c) = 0\). This means, if substituting a specific value into the polynomial results in zero, then the term \(x - c\) is a factor.

In practice, this theorem allows us to test possible roots of a polynomial through simple substitution. For example, if we have \(f(x)=x^4-6x^3-8x+48\) and suspect \(x-6\) might be a factor, we substitute \(x=6\). Finding that \(f(6) = 0\) confirms \(x-6\) is indeed a factor.

This theorem simplifies the often complex process of factorization by pinpointing the roots directly, saving us from trial and error approaches.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form \(x-c\), such as \(x-6\). It offers a more streamlined process than long division, especially when the leading coefficient of the divisor is 1.

To perform synthetic division, list the coefficients of the polynomial in order and then apply the value \(c\) from the divisor \(x-c\). For our problem, with \(f(x) = x^4 - 6x^3 - 8x + 48\), the coefficients are 1, -6, 0, -8, and 48. The value from \(x-6\) is \(6\).

The division involves multiplying down and adding across, resulting in a set of numbers that represent the quotient polynomial. This approach is not just efficient but also reduces computational complexity, making it easier to manage larger polynomials.

Long Division

Long division is another method used to divide polynomials. While it is less efficient than synthetic division for binomials, it is more versatile and applicable to a wider range of polynomials.

In long division, you set up the division as you would with numbers and solve for the quotient by iteratively dividing, multiplying and subtracting. When dividing \(x^4 - 6x^3 - 8x + 48\) by \(x-6\), you start by dividing the leading term of the dividend by the leading term of the divisor, then continue through the polynomial.

This method provides detailed insight into each step of the division process, making it excellent for learning purposes and cross-checking computations. While it may seem tedious, it is indispensable when dealing with divisors more complex than binomials.

Binomial Factor

Binomial factors are polynomials of the form \(x-c\), where \(c\) is a constant. They are fundamental building blocks in polynomial factorization and play a significant role in simplifying expressions.

Identifying binomial factors can significantly simplify the polynomial and help in solving polynomial equations. For instance, after verifying \(f(6) = 0\), we established that \(x-6\) is a binomial factor of \(f(x) = x^4-6x^3-8x+48\).

These factors often lead us to more hidden solutions of polynomial equations, facilitating further reduction into simpler expressions. Understanding binomial factors is crucial for mastering polynomial manipulation and solving.

Polynomial Roots

The roots of a polynomial are values of \(x\) that satisfy \(f(x) = 0\). They are solutions to the polynomial equation and can be visualized as points where the graph of the polynomial intersects the x-axis.

Finding polynomial roots is often the ultimate goal in factorizing a polynomial. Verified through Factor Theorem, synthetic or long division can further help to break down complex polynomials into manageable factors. For example, if \(x-6\) is a factor of the polynomial \(x^4-6x^3-8x+48\), then \(x=6\) is a root.

Knowing the roots helps in simplifying the polynomial fully and can lead to solving real-world problems modeled by polynomial equations. Understanding roots is key to linking algebraic solutions to graphical interpretations in mathematics.