Математика с д-р Атанас Илчев

1 Problem Find the numerical value of the expression $a^2+b^2$ if $a+b=5$ and $ab=9$. Solution: Let's recall the formula for truncated multiplication $(a+b)^2=a^2+2ab+b^2$, which we have already reviewed here. Notice that $a^2+b^2=(a+b)^2-2ab$, this is because $(a+b)^2-2ab=a^2+2ab+b^2-2ab$, and the ones complement $2ab$ and $-2ab$ are truncated. Thus, we have already figured out how to represent $a^2+b^2$ as the sum and product of $a$ and $b$, hence $a^2+b^2=(a+b)^2-2ab=5^2-2.9=25-18=7.$ Problem 2 Prove the identity $(ab+cd)^2=(a^2+c^2)(b^2+d^2)-(ad-cb)^2.$ Solution: We need to prove that the left side of this equality is equal to the right side. We will simplify both sides of the equality and compare them, hence $LS=a^2b^2+2abcd+c^2d^2$. Now we simplify the right-hand side of the equality  $LS=a^2b^2+a^2d^2+b^2c^2+c^2d^2-(a^2d^2-2adcb+c^2b^2)=$ $=a^2b^2+a^2d^2+b^2c^2+c^2d^2-a^2d^2+2adcb-c^2b^2=a^2b^2+c^2d^2+2abcd$.  Thus we get that $LS=RS$ and the equality is an identity...

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...