Problem 31 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 11 x^{3}-11 x+11, \quad 11 $$

Step-by-Step Solution

Answer

Answer: The other factor is $$x^3 - x + 1$$.

Step 1: Setup polynomial division

First, set up the polynomial division using the polynomial and the constant factor. $$\frac{11x^3 - 11x + 11}{11}$$

Step 2: Divide the polynomial by the constant factor

Now, we will divide each term of the polynomial by the constant factor. $$\frac{11x^3}{11} - \frac{11x}{11} + \frac{11}{11}$$

Step 3: Simplify the terms of the division

Simplify each term of the division by canceling out the common factors between the numerator and the denominator. $$x^3 - x + 1$$

Step 4: Write the result as the other factor

The simplified polynomial is the other factor we were looking for. The other factor is $$x^3 - x + 1$$.

Key Concepts

Polynomial Simplification

Polynomial simplification is all about reducing a complex polynomial to its simplest form. This process entails performing operations that remove any unnecessary components from the polynomial.
For instance, in our exercise, we have the polynomial \( 11x^3 - 11x + 11 \).
To simplify it, we began by dividing the entire polynomial by a constant factor, which in this case is \( 11 \).
Once you divide, you'll deal with each term individually:

• \( \frac{11x^3}{11} = x^3 \)
• \( \frac{-11x}{11} = -x \)
• \( \frac{11}{11} = 1 \)

By doing these calculations, we've removed the constant factor, leading to the simplest version of the polynomial: \( x^3 - x + 1 \).
Simplifying polynomials into their most reduced form makes subsequent mathematical operations significantly easier and clearer.

Factoring Polynomials

Factoring polynomials is about breaking down a polynomial into its divisible elements, or 'factors', just like numbers have factors.
A polynomial can often be factored into smaller, simpler polynomial components that multiply together to recreate the original polynomial.
Take our exercise, for example, where we're given the polynomial \( 11x^3 - 11x + 11 \) and asked to factor it out with \( 11 \).
To find the other factor, you divide the whole polynomial by \( 11 \). Doing so, as seen above, simplifies to \( x^3 - x + 1 \), which is the other factor.
Factoring is essential because it helps solve polynomial equations and greatly aids in understanding the polynomial's roots or zero points.
Always remember that factoring might not change the polynomial's value or properties; it just presents them in a more approachable format.

Monomial Division

Monomial division involves dividing a polynomial by a monomial – that is, a single-term polynomial.
In this context, dividing each term of a polynomial equally by a monomial simplifies the expression.
In our solution, we divided \( 11x^3 - 11x + 11 \) by the monomial \( 11 \). This means:

• Each term of the polynomial is separately divided by the monomial.
• Cancel out the common factors where possible in the division.

With our polynomial:

• Divide \( 11x^3 \) by \( 11 \) to get \( x^3 \).
• Divide \( -11x \) by \( 11 \) to get \( -x \).
• Divide \( 11 \) by \( 11 \) to get \( 1 \).

The result, \( x^3 - x + 1 \), stands as a simplified version of the original polynomial.
Dividing by monomials can hugely simplify problem-solving and is a great tool especially when dealing with higher degree polynomials.