Formula for the sum and difference of the cubes of two numbers - $(a\pm b)(a^2\mp ab+b^2)$

We continue with the next and last of the abbreviated multiplication formulas $(a\pm b)(a^2\mp ab+b^2)=a^3\pm b^3$. We will look at some problems to show some applications of it.

Problem 1 Perform the multiplication $(3-x)(9+3x+x^2).$

Solution: Notice that given the expression $(3-x)(9+3x+x^2)$, we can write it in the form $(3-x)(3^2+3x+x^2).$ We will apply the formula $(a-b)(a^2+ab+b^2)$, replacing $a$ with $3$ and $b$ with $x$, so we get $(3-x)(3^2+3x+x^2)=3^3-x^3.$

Problem 2 Perform the multiplication $(3t+2)(9t^2-6t+4).$

Solution: Given an expression, we can write it in the form $(3t+2)[(3t)^2-3t.2+2^2]$. Now it is easy to see that we can apply the formula $(a+b)(a^2-ab+b^2)$, where $a=3t$ and $b=2$, so we get $(3t+2)[(3t)^2-3t.2+2^2]=(3t)^3+2^3=27t^3+8.$

Problem 3 Simplify the expression $(x-2)(x^2+2x+4)-x(x-2)(x+2)-4(x-2).$

Solution: Apply the formulas $(a-b)(a^2+ab+b^2)=a^3-b^3$ and $(a-b)(a+b)=a^2-b^2$, hence $(x-2)(x^2+2x+4)-x(x-2)(x+2)-4(x-2)=x^3-2^3-x(x^2-4)-4x+8=x^3-8-x^3+4x-4x+8=0.$

4 Problem Simplify the expression and find its numerical value $(2y-3)(4y^2+6y+9)+(y+3)(y^2-3y+9),$ at $y=2$.

Solution: Now we calculate the numerical value of the expression at $y=2$, therefore the value of the expression is $9.2^3=72.$

5 Problem Find the unknown number $x$ in the equality $(x+4)(x^2-4x+16)-x(x-2)(x+2)=8.$

Solution: Apply the formulas $(a+b)(a^2-ab+b^2)=a^3+b^3$ and $(a-b)(a+b)=a^2-b^2$, from where we get $x^3+8-x(x^2-4)=8$, reveal the parentheses $x^3+8-x^3+4x=8$, from where we get that $4x=0$, therefore the number $x$ we are looking for is $0.$

6 Problem Find the numerical value of the expression $(x+2)(x^2-2x+4)-(2x+1)(2x-1)-9$ if $x=-2$

Solution: Apply the formulas $(a+b)(a^2-ab+b^2)=a^3+b^3$ and $(a-b)(a+b)=a^2-b^2$ with which you are already familiar from here, from which we get $x^3+2^3-[(2x)^2-1]-9=x^3+8-4x^2+1-9=x^3-4x^2$. Now we calculate the value of the expression for $x=-2$ and get $(-2)^3-4(-2)^2=-8-16=-24.$ 

7 Problem Prove that the value of the expression $(2x-1)(4x^2+2x+1)-8x(x+2)(x-2)-32x+4$ does not depend on the value of $x$.

Solution: Clearly, whatever $x$ is, this expression will always equal $3$.

8 Problem Prove the identity $(x-1)(x^2+x+1)-(x-1)^3=3x(x-1).$

Solution: We need to prove that the left side of this equality is equal to the right side. To do this, we will reveal the parentheses on the left side of the equality $LS=x^3-1-(x^3-3x^2+3x-1)=x^3-1-x^3+3x^2-3x+1=3x^2-3x.$ We also reveal the parentheses on the right side of the equality and get $3x(x-1)=3x^2-3x$. So we have obtained that the left side is equal to the right side and this equality is an identity.

Homework assignments

1. Perform the multiplication:

a) $(x-y)(x^2+xy+y^2);$ b) $(y^2-7)(y^4+7y^2+49);$ c) $(\frac{1}{2}a+b)(\frac{1}{4}a^2-\frac{1}{2}ab+b^2);$ d) $(\frac{1}{3}-2a)(\frac{1}{9}+\frac{2}{3}a+4a^2)$. 

2. Prove the identity: 

a) $(y-1)^3-(y+2)(y^2-2y+4)=3(-y^2+y-3);$ b) $(x^3+y^3)=(x+y)^3-3xy(x+y).$

3. Simplify the expression: 

a) $(3x-4)(9x^2+12x+16)-(3x-2)^3-2x(x+4);$ b) $(x-3)^2-(x-3)(x^2+3x+9).$

4. Find the numerical value of the expression: 

a) $3(x-1)^2+(x+2)(x^2-2x+4)-(x+1)^3$ at $x=\frac{1}{27};$

b) $-(b-1)(b^2+b+1)+b(b+3)(b-3)$ at $b=(-\frac{1}{9})^2;$

5. Given the expression $A=(x-m)^3-(x-m)(x^2+xm+m^2)-3mx(m+3)$, where $m$ is a parameter.

a) Reduce the expression to normal form.

b) For which value of $m$ is the coefficient of the second power term 18?

6. Given the expression $8+3(2y^2-y)-(y-3)(y^2+3y+9)+(y-2)^3:$

a) Find the normal form of the given expression;

b) Find the numerical value of the expression for $y=-3.$  

7. Given the expression $a-(a-b^2)(a^2+b^4+ab^2)+(a-b^3)(a+b^3):$

a) Find the normal form of the given expression;

b) Find the numerical value of the expression for $a=-2.$  

8. Given that $\frac{a+b}{a^3+b^3}=\frac{1}{6}$ and $a^2+b^2=10$, find how much $ab$ is equal to.$ 

9. If we have that $\frac{4(x^3-y^3)}{x^2+xy+y^2}+\frac{1}{2}(\frac{x^2-y^2}{x+y})=35$, how much will $x-y$ be equal to?

10. Calculate in a rational way the expression $\frac{68^2-68.32+32^2}{68^3+32^3}.$

11. Prove that the value of the expression $(-2y-1)^2-y(7y+1)-(y-1)^3+(y^2+y+1)(y-1)$ does not depend on the values of the variable $y$.

12. Show that no matter what is the value of the variable $z$ the expression 

$(z^2-2z+4)(z^2+2z+4)(z+2)(z-2)-(-64+z^6)$ is identically equal to $0$.