Problem 62 - Algebra 2

Question

Which binomial is NOT a factor of \(x^{3}-x^{2}-17 x-15 ?\) \(\begin{array}{llll}{\text { F. } x-5} & {\text { G. } x+1} & {\text { H. } x+3} & {\text { J. } x+5}\end{array}\)

Step-by-Step Solution

Answer

The binomial that is NOT a factor is the one for which substitution gives a non-zero remainder.

Step 1: Understand the problem

We need to determine which given binomial is NOT a factor of the polynomial \(x^{3}-x^{2}-17 x-15\). This can be done by applying the Remainder Theorem.

Step 2: Apply the Remainder Theorem for each option

According to the Remainder Theorem, if a binomial \(x - a\) is a factor of a given polynomial, then the remainder is 0 when we substitute \(a\) into the polynomial. We will substitute the value of \(-a\) from each option into the polynomial and check if the remainder is 0.

Step 3: Check each option

We evaluate the polynomial \(x^{3}-x^{2}-17 x-15\) at \(x = 5\), \(x = -1\), \(x = -3\), and \(x = -5\) for options F, G, H, and J, respectively.

Step 4: Determine the non-factor

If, after substitution, we get a remainder other than 0, that binomial is not a factor of the polynomial. We perform this check for each option.

Key Concepts

Remainder Theorem

The Remainder Theorem is a vital concept in algebra, especially when dealing with polynomials. It states that if a polynomial function is divided by a binomial of the form \(x - a\), the remainder of this division is the value of the polynomial function at \(x = a\). In simpler terms, to check if a given binomial is a factor of a polynomial, plug the value \(a\) into the polynomial. If the result equals zero, the binomial \(x - a\) is indeed a factor. This theorem is particularly handy when you need to quickly determine factors without performing long division.

For instance, if we are examining whether \(x - 5\) is a factor of a polynomial \(P(x)\), evaluate \(P(5)\). If \(P(5) = 0\), \(x - 5\) is a factor. This theorem saves time and effort and simplifies the process of factorization significantly. When we test each option provided in the exercise using the Remainder Theorem, we determine the appropriate factor by seeing which substitution yields a non-zero remainder.

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial for a specific value of its variable. This is essentially what we are doing when applying the Remainder Theorem—you are evaluating the polynomial at a specific point to determine the remainder. For the polynomial \(x^{3} - x^{2} - 17x - 15\), we substitute in the values of \(x\) from the given options. \

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• For \(x - 5\), evaluate the polynomial at \(x = 5\).\
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• For \(x + 1\), evaluate at \(x = -1\).\
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• For \(x + 3\), evaluate at \(x = -3\).\
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• For \(x + 5\), evaluate at \(x = -5\).\
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During evaluation, make sure each term of the polynomial is correctly calculated—be attentive to sign changes and exponents. The correct evaluation will reveal the non-factor if there exists any, as it will result in a non-zero value.

Binomial Factors

Binomial factors are factors of a polynomial that are made up of two terms, hence 'bi-' meaning 'two', and '-nomial' as in 'terms'. Binomial factors generally look like \(x - a\) or \(x + b\), where \(a\) and \(b\) are constants. The significance of finding binomial factors lies in simplifying polynomials and trying to solve them, for example, by finding zeros or graphing.

To find if a binomial is a factor of a given polynomial, one can factor the polynomial completely or use the Remainder Theorem as a shortcut. When using the Remainder Theorem, if substituting \(a\) into the polynomial gives a remainder of zero, then \(x - a\) is a factor; if not, it's not a factor. This direct evaluation allows us to bypass more complex factorization methods and is particularly useful for higher degree polynomials where direct factorization is not straightforward. In the exercise provided, this concept allowed us to test the possible binomial factors by simple substitution, revealing which binomial does not factor into the given polynomial.