Problem 20 - Algebra 2

Question

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{3}+x^{2}-16 x-16 ; x+4 $$

Step-by-Step Solution

Answer

The remaining factors are \(x - 4\) and \(x + 1\).

Step 1: Divide the Polynomial by the Given Factor

Given the polynomial \(x^3 + x^2 - 16x - 16\) and the factor \(x+4\), we use polynomial division to find the quotient. Divide \(x^3 + x^2 - 16x - 16\) by \(x+4\). Start by dividing the first term of the dividend \(x^3\) by the first term of the divisor \(x\), which gives \(x^2\). Multiply \(x^2\) by \(x+4\), resulting in \(x^3 + 4x^2\). Subtract \(x^3 + 4x^2\) from the original polynomial to get \(-3x^2 - 16x - 16\).

Step 2: Continue Polynomial Division

Continuing from \(-3x^2 - 16x - 16\), divide \(-3x^2\) by \(x\) to get \(-3x\). Multiply \(-3x\) by \(x+4\) to get \(-3x^2 - 12x\). Subtract \(-3x^2 - 12x\) from \(-3x^2 - 16x - 16\) to get \(-4x - 16\).

Step 3: Finish the Division

Now divide \(-4x\) by \(x\) to obtain \(-4\). Multiply \(-4\) by \(x+4\) to produce \(-4x - 16\). Subtract \(-4x - 16\) from \(-4x - 16\), yielding a remainder of zero. The quotient we have found during our division steps is \(x^2 - 3x - 4\).

Step 4: Factor the Quotient

Now, factor the quadratic \(x^2 - 3x - 4\). Look for two numbers that multiply to \(-4\) and add up to \(-3\). These numbers are \(-4\) and \(1\). So, the quadratic can be factored into \((x - 4)(x + 1)\).

Step 5: List All Factors

We found that \(x^2 - 3x - 4 = (x - 4)(x + 1)\). Therefore, the complete factorization of the original polynomial is \((x + 4)(x - 4)(x + 1)\). So, the remaining factors of the polynomial are \(x - 4\) and \(x + 1\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex polynomial into simpler polynomials that, when multiplied together, yield the original polynomial. It's like decoding the polynomial into smaller, manageable pieces. This process helps in simplifying polynomial expressions and solving polynomial equations.

• First, identify any obvious factors or common terms in the polynomial.
• Then, apply methods such as synthetic division, long division, or factoring by grouping to find other factors.
• Check your work by multiplying the factors to see if you get the original polynomial.

Factoring is a crucial skill that provides insights into the roots and behavior of polynomials. For instance, if given a polynomial like \(x^3 + x^2 - 16x - 16\) and a factor \(x+4\), polynomial division will help find other factors by simplifying the expression further.

Quadratic Factorization

Quadratic factorization is a specific type of factoring focused on quadratic equations, which are polynomials of degree two. The typical form is \(ax^2 + bx + c\). The goal is to express this quadratic as a product of two binomials.

• The key here is to find two numbers that multiply to \(c\) (the constant term) and add up to \(b\) (the coefficient of the middle term).
• Once these numbers are determined, you can express the quadratic as \((x + p)(x + q)\), where \(p\) and \(q\) are the numbers found.

In our solution, the quadratic \(x^2 - 3x - 4\) can be factored into \((x - 4)(x + 1)\), with \(-4\) and \(+1\) being the key numbers. This simplifies solving and analyzing quadratic equations.

Remainder Theorem

The Remainder Theorem is a useful concept in polynomial division. It states that when a polynomial \(f(x)\) is divided by a linear divisor \(x - c\), the remainder of the division is \(f(c)\). This theorem allows us to quickly test whether a linear polynomial is a factor of another polynomial.

• If \(f(c) = 0\), then \(x - c\) is a factor of the polynomial \(f(x)\).
• If there is a non-zero remainder when \(x - c\) divides \(f(x)\), then \(x - c\) is not a factor.

In our division process, obtaining a remainder of zero after dividing \(x^3 + x^2 - 16x - 16\) by \(x + 4\) confirms that \(x + 4\) is indeed a factor. The Remainder Theorem streamlines the factor-finding process, saving time and effort by verifying potential factors efficiently.