Problem 36 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ 48 v^{3}+56 v^{2}+32 v $$

Step-by-Step Solution

Answer

8v(6v^2 + 7v + 4) is the factored form. 6v^2 + 7v + 4 is a prime polynomial.

Step 1 Title - Find the Greatest Common Factor (GCF)

Identify the common factor in each term of the polynomial. The terms are: 48v^3, 56v^2, and 32v. The GCF of the coefficients is 8, and since each term has at least one v, the GCF of the entire polynomial is 8v.

Step 2 Title - Factor Out the GCF

Factor out the GCF from the polynomial. Divide each term by the GCF (8v) to rewrite the polynomial as 8v(6v^2 + 7v + 4).

Step 3 Title - Check Your Work

To ensure the factoring is correct, distribute 8v back into each term of the polynomial inside the parenthesis. This should give you back the original polynomial: 8v(6v^2) + 8v(7v) + 8v(4) = 48v^3 + 56v^2 + 32v.

Step 4 Title - Identify Prime Polynomials

Review the polynomial inside the parenthesis: 6v^2 + 7v + 4. Since it can’t be factored further using integer coefficients, it is a prime polynomial.

Key Concepts

Greatest Common Factor

When factoring polynomials, the first step is to find the Greatest Common Factor (GCF). The GCF is the highest number that divides all coefficients of the polynomial without leaving a remainder. It also includes the highest power of any variables common to each term.
To find the GCF of the polynomial:

• Look at the coefficients of each term. For example, in 48v^3, 56v^2, and 32v, the GCF of 48, 56, and 32 is 8.
• Next, identify the common variable factor. Here, each term has at least one 'v', so the GCF of the whole polynomial is '8v'.
• Thus, for the given polynomial 48v^3 + 56v^2 + 32v, the GCF is 8v.

Prime Polynomials

A prime polynomial is a polynomial that can't be factored into the product of two non-constant polynomials with integer coefficients. Once you've factored out the GCF from a polynomial, you need to check if the remaining polynomial can be factored further.
For example, after factoring out 8v from 48v^3 + 56v^2 + 32v, you get 8v(6v^2 + 7v + 4). Now, inspect the polynomial within the parenthesis:

• Check if 6v^2 + 7v + 4 can be factored into smaller polynomials or if it is prime.
• Since 6v^2 + 7v + 4 cannot be factored using integer coefficients, it is a prime polynomial. Thus, no further factorization is possible.

Polynomial Division

Polynomial division is a method used to divide one polynomial by another polynomial of lesser or equal degree. This is important when simplifying expressions or solving polynomial equations.
Here is a simple walkthrough on polynomial division:

• First, ensure the polynomial you're dividing by (the divisor) is simpler than the original polynomial.
• Next, factor out the GCF from the original polynomial to simplify it. For example, in this exercise: \(48v^3 + 56v^2 + 32v = 8v(6v^2 + 7v + 4)\).
• Align the polynomial terms in descending order of degree for simplicity.
• Then, divide each term in the original polynomial by the corresponding term in the divisor.

Finally, double-check your division by multiplying the divisor by your result. This product should equal the original polynomial, confirming the correctness of your factorization and division.