Problem 117 - Introductory Algebra for College Students

Question

Write a polynomial that fits the given description. Do not use a polynomial that appears in this section or in the Exercise Set. The polynomial has four terms and can be factored by grouping.

Step-by-Step Solution

Answer

The polynomial needed is \( ax^3 + bx^2 + ax + b \), where \( a \) and \( b \) are any constants.

Step 1: Write a polynomial with four terms

We start by writing any polynomial of four terms. We do not worry about the factorization at this step, just about making sure we have exactly four terms. A possible polynomial could be \( ax^3 + bx^2 + ax + b \).

Step 2: Factor the polynomial by grouping

We now attempt to factor the polynomial by grouping. This technique involves splitting the polynomial into two pairs and factoring out the GCF of each pair. Our polynomial can be grouped as follows: \( ax^3 + bx^2 + ax + b = (ax^3 + bx^2) + (ax + b) \). Now, we can factor out the GCF of each group, resulting in \( ax^2(x + b) + 1(x + b) \). We can see that the polynomial can indeed be factored by grouping.