Question

Mikisiti Saila \((1939-2008),\) an Inuit artist from Cape Dorset, Nunavut, was the son of famous soapstone carver Pauta Saila. Mikisita's preferred theme was wildlife presented in a minimal but graceful and elegant style. Suppose a carving is created from a rectangular block of soapstone whose volume, \(V\), in cubic centimetres, can be modeled by \(V(x)=x^{3}+5 x^{2}-2 x-24\) What are the possible dimensions of the block, in centimetres, in terms of binomials of x.

Short answer

Possible dimensions are \(x + 3\), \(x + 4\), and \(x - 2\).

Step-by-step solution

Identify the problem

Given the volume of a rectangular block of soapstone modeled by the polynomial function \(V(x) = x^3 + 5x^2 - 2x - 24\), it is necessary to find the possible binomial factors of this polynomial.

Apply the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial is a factor of the constant term (-24) divided by a factor of the leading coefficient (1). Thus, the potential rational roots of \(x^3 + 5x^2 - 2x - 24\) are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\).

Test the possible roots

Test these potential roots using synthetic division or direct substitution. For example, test \(x = -3\):\[ V(-3) = (-3)^3 + 5(-3)^2 - 2(-3) - 24 = -27 + 45 + 6 - 24 = 0 \]Since \(x = -3\) is a root, \(x + 3\) is a factor.

Perform polynomial division

Perform polynomial division of \(x^3 + 5x^2 - 2x - 24\) by \(x + 3\). This helps to factor the polynomial further:\[ \frac{x^3 + 5x^2 - 2x - 24}{x + 3} = x^2 + 2x - 8 \]So, \( V(x) = (x + 3)(x^2 + 2x - 8) \).

Factor the quadratic expression

Factor the quadratic \(x^2 + 2x - 8\) further:\[ x^2 + 2x - 8 = (x + 4)(x - 2) \]Thus, \( V(x) = (x + 3)(x + 4)(x - 2) \).

State the possible dimensions

The possible dimensions of the block of soapstone, in centimetres, are represented by the binomials \(x + 3\), \(x + 4\), and \(x - 2\).

Key concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool in polynomial factoring. It helps us identify potential rational roots (solutions) of a polynomial equation. This theorem states that any potential rational root of a polynomial function is a factor of the constant term divided by a factor of the leading coefficient.
For example, in the polynomial function \(V(x) = x^3 + 5x^2 - 2x - 24\), the constant term is -24, and the leading coefficient is 1. Thus, the potential rational roots are the factors of -24 divided by 1, which give us the set \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\).
This step significantly narrows down the list of potential roots that can be tested in subsequent steps.

polynomial division

Once you have identified a potential rational root using the Rational Root Theorem and have verified it as a true root by substituting it back into the polynomial, the next step is to perform polynomial division. Polynomial division helps to break the polynomial into simpler factors.
In our example, we found that \(x = -3\) is a root of the polynomial \(V(x) = x^3 + 5x^2 - 2x - 24\). So, we divide \(V(x)\) by \((x + 3)\). Using polynomial long division or synthetic division, we get:\[ \frac{x^3 + 5x^2 - 2x - 24}{x + 3} = x^2 + 2x - 8 \]
This quotient is a simpler polynomial that we will factor further.

quadratic factorization

Quadratic factorization is an essential skill for breaking down polynomials into simpler binomial factors. When you have a quadratic polynomial, like the quotient \(x^2 + 2x - 8\) obtained from polynomial division, you can factor it further.
The quadratic polynomial \(x^2 + 2x - 8\) can be factored into:\[ x^2 + 2x - 8 = (x + 4)(x - 2) \]
We look for two numbers that multiply to the constant term (-8) and add up to the coefficient of the linear term (+2). In this case, those numbers are +4 and -2.

binomial factors

Binomial factors are the simplest building blocks of a polynomial. After you have factored a polynomial completely, it should be expressed as a product of binomial factors.
In our ongoing example, we have completely factored the polynomial \(V(x) = x^3 + 5x^2 - 2x - 24\) into its binomial factors:\[ V(x) = (x + 3)(x + 4)(x - 2) \]
Each of these binomials represents one dimension of the block of soapstone. In centimeters, the possible dimensions of the soapstone block can be given by the binomials \(x + 3\), \(x + 4\), and \(x - 2\).
Understanding binomial factors is critical because they are the simplest and most straightforward expressions that we work with when solving polynomial equations.