Decomposing a polynomial into factors by applying the formulas for short multiplication

 Each of the abbreviated multiplication formulas we have looked at so far is an example of representing a polynomial as a product, for example $a^2-b^2=(a-b)(a+b)$, on the left hand side we have the polynomial $a^2-b^2$ and on the right hand side the product of factors $(a-b)(a+b)$. In the same way, $a^2\pm 2ab+b^2=(a\pm b)^2$. Clearly we see that again we have a polynomial on the left hand side of the equality and a product on the right hand side. Let's write down the other formulas with the left and right parts swapped $a^3\pm 3a^2b+3ab^2\pm b^3=(a\pm b)^3$ and $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2).$ We'll see below how we can apply these equalities to specific problems.

Problem 1 Decompose the polynomial $4x^2-y^2$ into factors. 

Solution: In our case $a=2x$ and $b=y$, therefore $4x^2-y^2=(2x-y)(2x+y).$

Problem 2 Decompose the polynomial $p^2-(x+y)^2$ into factors.$

Solution: Again apply $a^2-b^2=(a-b)(a+b)$, in this case $a=p$, $b=x+y$, hence $[p-(x+y)](p+x+y)=(p-x-y)(p+x+y)$.

3 Problem Decompose the polynomial $16b^2-8b+1$ into factors.

Solution: To decompose this polynomial, we will apply the formula $(a-b)^2=a^2-2ab+b^2$ that we considered here, taken "from left to right", i.e. $a^2-2ab+b^2=(a-b)^2.$

4 Problem Decompose the polynomial $8x^3+12x^2y+6xy^2+y^3$ into factors. 

Solution: Let us write the given polynomial in the form $(2x)^3+3.(2x)^2.y+3.2x.y^2+y^3.$ It is not hard to notice that we can apply the formula $a^3+ 3a^2b+3ab^2+b^3=(a+b)^3$ (more about this formula can be found here), in our case $a=2x$ and $b=y$, therefore $(2x)^3+3.(2x)^2.y+3.2x.y^2+y^3=(2x+y)^3.$

5 Problem Decompose the polynomial $64a^3+27b^3.$ into factors 

Solution: Notice that given polynomial, we can represent in the form $(4a)^3+(3b^3).$ Apply the formula $a^3+b^3=(a+b)(a^2-ab+b^2)$ (more about this formula can be seen here), from where we get that $(4a)^3+(3b^3)=(4a+3b)(16a^2-12ab+9b^2).$

6 Problem Decompose the triplet $x^2-8x+15$ into factors.

Solution: Now it is not difficult to see that $x^2-2.x.4+16=(x-4)^2$ (a^2-2ab+b^2=(a-b)^2), hence $x^2-8x+15=(x-4)^2-1$. We still have one more step to solve the problem. Let us now consider $(x-4)^2-1=(x-4)^2-1^2$. We apply the formula $a^2-b^2=(a-b)(a+b)$, where $a=x-4$ and $b=1$ and obtain that $(x-4)^2-1=(x-4)^2-1^2=(x-4-1)(x-4+1)=(x-5)(x-3).$

7 Problem Find the largest value of $\frac{6}{x^2+4x+5}$ and the value of $x$ at which it is obtained.

Solution: represent the denominator of the given fraction in the form $\frac{6}{x^2+2.x.2+2^2+1}$, hence $\frac{6}{x^2+2.x.2+2^2+1}=\frac{6}{(x+2)^2+1}$. Clearly, the smaller the denominator of a fraction, the larger it is (it's one thing to divide a pizza among three, it's another if you divide it among six). The smallest value that $(x+2)^2$ can take is $0$, since $(x+2)^2\geq 0$ for any $x$, with $(x+2)^2=0$ when $x=-2$. From here we conclude that the largest value that the given fraction can take is at $x=-2$ and it is equal to $\frac{6}{0+1}=6.$

8 Problem To decompose into factors the polynomial $x^6-1.$

Solution: Now apply the formula $a^2-b^2=(a-b)(a+b)$, where $a=x^3$ and $b=1$, hence $(x^3)^2-1^2=(x^3-1)(x^3+1)$. For the first multiplier we use that $a^3-b^3=(a-b)(a^2+ab+b^2)$, and for the second $a^3+b^3=(a+b)(a^2-ab+b^2)$, where $(x^3-1)(x^3+1)=(x-1)(x^2+x+1)(x+1)(x^2-x+1)$, which solves the problem. Try decomposing this polynomial using that $x^6-1=(x^2)^3-1^3.$

9 Problem Decompose the polynomial $x^{2k}+2x^ky^l+y^{2l}$ into factors.

Solution: We can represent the polynomial in the form $(x^k)^2+2x^ky^l+(y^l)^2$. Now substituting in $a^2+2ab+b^2=(a+b)^2$ with $a=x^k$ and $b=y^l$, we get $x^{2k}+2x^ky^l+y^{2l}=(x^k+y^l)^2.$

Homework assignments

1. Decompose the polynomial into factors:

a) $x^2+18x+81;$ b) $4x^2-16y^2;$ c) $y^3+15y^2+75y+125;$ d) $27u^3-64v^3.$ 

2. Find the numerical value of the expression $(2x-1)^2-81x^2$ when $x=1.$

3. Prove that if $a+b$ is divisible by $5$, then $a^2-b^2$ is also divisible by 5.

4. Find the smallest value of the expression $x^2-10x+37$ and the value of $x$ at which it is reached.

5. Prove the identity $(a(x+y)+b(x-y))^2-(a(x-y)+b(x+y))^2=4xy(a-b)(a+b)$.