Decomposition of a polynomial by exporting a common factor

Often in solving various problems we have to represent a polynomial as a product of factors. These multipliers can be either monomials or other polynomials. For example, it is well known that $a(b+c)=a.b+a.c$. If we write this equality in reverse order i.e. $a.b+a.c=a(b+c)$ we see that we have already represented the sum of the monomials $ab$ and $bc$ as a product. Let us consider some problems.

1 Problem Calculate rationally $14.85+14.15.$

Solution: The first warrant for solving this problem is to first calculate the product $14.85$ and the product $14.15$ and then add the resulting numbers. The way is not wrong of course, but it does not fit the word rational. Let us now instead consider the equality $a.b+a.c=a(b+c)$, where $a=14$, $b=85$ and $c=15$. We replace the letters with their corresponding equal numbers and get $14.85+14.15=14.(85+15)=14.100=1400.$ 

Problem 2 Decompose the polynomial $5t+5m$ into factors.

Solution: Notice that in the first and second addends we have the same factor $5$, therefore we can apply $a.b+a.c=a(b+c)$, i.e. $5t+5m=5(t+m).$

Problem 3 Decompose the polynomial $3x-3y^4$ into factors.

Solution: The identity factor in this case is $3$, so by putting it in front of parentheses we get $3(x-y^4).$

4 Problem Decompose the polynomial $20x^2y^3+25x^3y^4+30x^4y^3$ into factors. 

Solution: Notice that the factors of all the monomials involved in this polynomial can be divided by 5, i.e. we can output 5 as a common factor. Moreover, each of the monomials in this polynomial contains the variables $x$ and $y$ of different degree. We take the factor 5, and the lowest powers of the variables $x$ and $y$ in the case $x^2y^3$ in front of parentheses, hence $20x^2y^3+25x^3y^4+30x^4y^3=5x^2y^3(4+5xy+6x^2).$

5 Problem Decompose the polynomial $2(2y-5)-3y(2y-5)$ into factors.

Solution: Notice that in the first and second products we have the same multiplier $(2y-5)$, therefore we can put it in front of parentheses and so we get $2(2y-5)-3y(2y-5)=(2y-5)(2-3y).$

6 Problem Decompose the polynomial $x(x+2y)+y(x+2y)$ into factors.

Solution: Here the common factor is $x+2y$, therefore $x(x+2y)+y(x+2y)=(x+2y)(x+y).$

7 Problem Decompose the polynomial $xy+3xy^2$ into factors.

Solution: The given polynomial can be written in the form $1.x.y+3x.y.y.$ The common factor in this case is $xy$, hence $xy(1+3y)$. Let us note that in the literature only $xy$ is written instead of $1.xy$.

8 Problem Decompose into factors the polynomial $4m(5-n)-3t(n-5).$

Solution. However, there is a way to make them the same, let's recall that $(n-5)=-(5-n)$, of course the check can easily be done. Therefore, to change the places of the addends I need to put a "-" sign in front of the parentheses. Now we are ready to decompose $4m(5-n)-3t(n-5)=4m(5-n)-3t[-(5-n)]=4m(5-n)+3t(5-n)=(5-n)(4m+3t).$ 

9 Problem Decompose the polynomial $3x(x-y)-5(x-y)^2$ into factors.

Solution: Express the common factor $(x-y)$ in front of parentheses, hence $3x(x-y)-5(x-y)^2=(x-y)[3x-5(x-y)]=(x-y)(3x-5x+5y)=(x-y)(-2x+5y).$

10 Problem Calculate the value of the expression $12m(4m-3)-3n(4m-3)-(3-4m)$ at $m=10$ and $n=7.$

Solution: The common factor in this polynomial is $(4m-3)$, of course in the last bracket

$(3-4m)=-(4m-3)$, следователно $12m(4m-3)-3n(4m-3)-(3-4m)=12m(4m-3)-3n(4m-3)+1.(4m-3)=(4m-3)(12m-3n+1)$. Now substituting $m=10$ and $n=7$ into the resulting product, we get that the numerical value of the expression is $(4.10-3)(12.10-3.7+1)=37.100=3700.$

Homework assignments

1. Decompose the polynomial into factors:

a) $3mb+m;$ b) $\frac{5}{3}-\frac{5}{3}x;$ c) $15x^2y+30xy^3;$ d) $36a^2b^2-27b^3;$ e) $5t^7+t^11.$ 

2. Decompose the polynomial into factors:

a) $7(x-z)+p(x-z);$ b) $2x(a+3b)-3y(a+3b);$ c) $(m-3)5+(m-3)3n;$ d) $(k+p)-2x(k+p);$ e) $3x(5y-7z)+7t(5y-7z).$ 

3. Decompose the polynomial into factors:

a) $2(x-y)+3z(x-y)+4q(y-x);$ b) $5(a+b)^2-c(a+b);$ c) $2(p-q)^3-3m(p-q);$ d) $x+y+3(x+y);$ e) $3(p-q)+7x(p-q)^2+4y(q-p)^3;$ f) $2a(2x-5)+b(5-2x).$ 

4. Calculate the value of the expression $x^6y^2+x^3y^3-x^5y^4$, for $x=1$ and $y=2.$

5.  Decompose the polynomial into factors:

a) $2a^m+3a^{m+1}+4a^{m+2};$ b) $(a+b+c)^2-na-nb-nc.$