Problem 60 - Elementary Algebra

Question

(a) factor by grouping. Identify any prime polynomials. (b) check. $$ 33 x z-22 a x-21 y z+14 a y $$

Step-by-Step Solution

Answer

Factorized polynomial: \((33 z - 22 a) (x - 7 y)\). Verified by multiplying back.

Step 1 - Group terms

Group the polynomial into two pairs: Let's take the given polynomial: \(33 x z - 22 a x - 21 y z + 14 a y\). Group terms as follows: Group 1: \(33 x z - 22 a x\) Group 2: \(-21 y z + 14 a y\).

Step 2 - Factor out the greatest common factor (GCF)

Extract the GCF from each group: From Group 1: \(33 x z - 22 a x = x (33 z - 22 a)\) From Group 2: \(-21 y z + 14 a y = -7 y (3 z - 2 a)\)

Step 3 - Combine like factors

Notice that both groups now contain a common factor \((33 z - 22 a)\): So, we can factor out \((33 z - 22 a)\): \(x (33 z - 22 a) - 7 y (33 z - 22 a) = (33 z - 22 a) (x - 7 y)\) Thus, \(33 x z - 22 a x - 21 y z + 14 a y = (33 z - 22 a) (x - 7 y)\)

Step 4 - Check the factorization

Multiply the factors back together to ensure they give the original polynomial: \((33 z - 22 a) (x - 7 y) = 33 z x - 231 y z - 22 a x + 154 a y\) This matches the original polynomial, verifying the factorization is correct.

Key Concepts

factoring polynomials

Factoring polynomials is a common algebraic process where we express a polynomial in terms of products of polynomials. This breaks down the original polynomial into simpler factors, making it easier to solve or simplify.
Let's take a quick look at an example. Given the polynomial:
33xz - 22ax - 21yz + 14ay,
You can rewrite it by grouping:
Group 1: 33xz - 22ax
Group 2: -21yz + 14ay.
Once grouped, we will use factoring by grouping to simplify it further.

greatest common factor

When factoring by grouping, the greatest common factor (GCF) plays a critical role. The GCF is the highest number or variable that divides two or more numbers or terms without any remainder. By finding the GCF, we can simplify each group in our polynomial.
Let's find the GCF for each group in our example:
For Group 1 (33xz - 22ax), the GCF is 'x':
Group 1: 33xz - 22ax = x(33z - 22a).
For Group 2 (-21yz + 14ay), the GCF is '-7y':
Group 2: -21yz + 14ay = -7y(3z - 2a).
Factoring out these GCFs simplifies our polynomial.

polynomial factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. This helps in solving equations or simplifying expressions.
In our example, after extracting the GCFs, we have:
x(33z - 22a) - 7y(33z - 22a).
Notice that (33z - 22a) is common in both terms.
We can factor it out as a single entity:
(33z - 22a)(x - 7y).
This is our factored polynomial.

check factorization

Checking factorization is essential to ensure the process was done correctly. We do this by multiplying the factors back together and verifying if we get the original polynomial.
Let's check our factorization:
Multiply (33z - 22a) by (x - 7y):
(33z - 22a)(x - 7y) = 33zx - 231yz - 22ax + 154ay.
We see that this product matches our original polynomial:
33xz - 22ax - 21yz + 14ay.
This confirms our factorization is correct and complete.