
Factorize the given polynomial : \[15pq + 15 + 9q + 25p\]
Answer
452.1k+ views
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.
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.
Recently Updated Pages
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Difference Between Plant Cell and Animal Cell

What is pollution? How many types of pollution? Define it

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.
