Question

Factorize the given polynomial : \[15pq + 15 + 9q + 25p\]

Answer 1

Hint: We will rearrange the terms in the given polynomial in order to take out common factors. Then, we will find the factors of the polynomial. Factorization or factoring is defined as the breaking or decomposition of an entity which may be a number, a matrix, or a polynomial into a product of another entity, or factors, which when multiplied together give the original number or a matrix.

Complete step-by-step answer:
The given polynomial is \[15pq + 15 + 9q + 25p\].
Let us rearrange the terms of the polynomial so that we have two terms with one common factor and another two terms with some other common factor.
In the given polynomial we see that the terms \[15pq\] and \[9q\] have common factors.
Let us factorize each term and check which factors are common.
Now, \[15pq\] can be written as:
\[15pq = 5 \times 3 \times p \times q\]
\[9q\] can be written as:
\[3 \times 3 \times q\].
We observe that in the terms \[15pq\] and \[9q\], the factors \[3\] and \[q\] are common i.e., \[3q\] is a common factor.
Hence, we will take \[3q\] out as a common factor from \[15pq\] and \[9q\].
Therefore, we have
\[15pq + 9q = 3q(5p + 3)\] ………\[(1)\]
Now, the terms remaining are \[15\] and \[25p\].
Let us check the common factors of these terms.
We can write \[15\] as \[15 = 3 \times 5\].
Also, \[25p\] can be written as:
\[25p = 5 \times 5 \times p\].
We observe that in both terms the common factor is \[5\]. So, we will take \[5\] common out. Thus, we get
\[15 + 25p = 5(3 + 5p)\] ……….\[(2)\]
Hence, we have expressed the given polynomial as follows:
\[15pq + 15 + 9q + 25p = (15pq + 9q) + (15 + 25p)\] ……….\[(3)\]
Using equations \[(1)\], and \[(2)\] in equation \[(3)\], we get
\[15pq + 15 + 9q + 25p = 3q(5p + 3) + 5(3 + 5p)\] ……….\[(4)\]
Since addition is commutative, \[5p + 3\] is the same as \[3 + 5p\]. We observe in equation \[(4)\], that on the RHS, the term \[5p + 3\] is common.
Factoring common terms, we get
\[15pq + 15 + 9q + 25p = (5p + 3)(3q + 5)\]

Note: Factoring Polynomials by Grouping method is also said to be factoring by pairs. Here, the given polynomial is distributed in pairs or grouped in pairs to find the zeros. The factorization can be done also by using algebraic identities. We should know the importance of factoring. Factoring is an important process that helps us understand more about our equations. Through factoring, we rewrite our polynomials in a simpler form, and when we apply the principles of factoring to equations, we yield a lot of useful information.