Problem 63 - Algebra 2

Question

Which of the following, when multiplied by \(x-1,\) results in a cubic polynomial whose standard form has three terms? \(\begin{array}{llll}{\text { A. }(x-1)^{2}} & {\text { B. } x^{2}-x} & {\text { C. } x^{2}-1} & {\text { D. } x-1}\end{array}\)

Step-by-Step Solution

Answer

Option B is correct, resulting in the cubic polynomial \(x^3 - 2x^2 + x\).

Step 1: Understanding the Question

We need to find an option that, when multiplied by \(x-1\), gives us a cubic polynomial with exactly three terms.

Step 2: Multiplying Each Option

Check each option by multiplying it with \(x-1\).

Step 3: Option A: \((x-1)^2\)

Multiplying \((x-1)^2\) by \(x-1\) results in \((x-1)^3\), which expands to \(x^3 - 3x^2 + 3x - 1\). This has four terms, not three.

Step 4: Option B: \(x^2-x\)

Multiplying \(x^2-x\) by \(x-1\) results in \(x^3-x^2-x^2+x\), which simplifies to \(x^3 - 2x^2 + x\). This has three terms and is a cubic, so this could be the answer.

Step 5: Option C: \(x^2-1\)

Multiplying \(x^2-1\) by \(x-1\) results in \(x^3-x^2-x+1\), which simplifies to \(x^3 - x^2 - x + 1\). This has four terms, not three.

Step 6: Option D: \(x-1\)

Multiplying \(x-1\) by \(x-1\) results in \((x-1)^2\), which is not a cubic polynomial.

Step 7: Identifying the Correct Option

Only Option B, when multiplied by \(x-1\), results in a cubic polynomial with exactly three terms: \(x^3 - 2x^2 + x\).

Key Concepts

Polynomial Multiplication

When we talk about polynomial multiplication, we refer to the process of multiplying two polynomials to find their product. It is a basic algebraic operation that requires you to distribute each term in the first polynomial to every term in the second polynomial.

Let's imagine we have two polynomials, A and B. For the multiplication A x B to be performed, we use the distributive property: each term of polynomial A is multiplied by each term of polynomial B. For example, if we are given A = a + b and B = c + d, the result after multiplying them would be (a + b)(c + d) = ac + ad + bc + bd.

In our exercise, we multiply each option by x - 1 to see which one yields a cubic polynomial with three terms. Multiplying polynomials systematically can help ensure that no terms are missed and the expansion is complete.

Factoring Polynomials

The process of factoring polynomials is the reverse of polynomial multiplication. It involves breaking down a complex polynomial into simpler 'factors' that, when multiplied together, give back the original polynomial. Factors are usually polynomials of lower degrees that are multiplied together to obtain the original polynomial.

Why is Factoring Important?

Factoring is a powerful tool in algebra because it simplifies equations and can make solving them much easier. For example, when finding zeros of polynomials or simplifying fractions that contain polynomials, factoring can be extremely useful.

In relation to our exercise, if we were given the result and asked to factor it to obtain one of the original binomial factors, we would be engaging in factoring polynomials. Here, however, our task is to find the product that gives us the desired form of a cubic polynomial.

Standard Form of a Polynomial

The standard form of a polynomial is when the polynomial is written in descending order of its terms based on the power of its variables. For instance, a cubic polynomial in standard form with one variable x should look like ax^3 + bx^2 + cx + d, where a, b, c, and d are constants, and a is nonzero.

Why does the standard form matter? It allows us to quickly identify the degree of the polynomial, which is indicated by the highest power of x, and it also makes it easier to compare and perform operations on polynomials.

Returning to our exercise, we look for the option that forms a cubic polynomial with three terms after multiplication with x - 1. A cubic polynomial with three terms, like the result we got from Option B, x^3 - 2x^2 + x, is already in standard form: the powers of x are in descending order and there are no like terms to combine.