cadaVR anatomy by Mozaik Education

Special algebraic products

Square of a binomial sum

\latex{(a+b)^2}=\latex{(a+b)(a+b)}=\latex{a^2+ab+ba+b^2}=\latex{a^2+2ab+b^2}

\latex{(a+b)^2=a^2+2ab+b^2}

The square of a binomial sum equals to the sum of the following: the square of the first term, twice the product of the two terms and the square of the second term.

E.g.

\latex{(x+3)^2=x^2+2\times x\times3+3^2=x^2+6x+9;\\[10pt]}

\latex{(2a+5)^2=(2a)^2+2\times2a\times5+5^2=4a^2+20a+25;\\[10pt]}
\latex{(3a+4b)^2=(3a)^2+2\times3a\times4b+(4b)^2=9a^2+24ab+16b^2;\\[10pt]}

\latex{\left( \frac{1}{2}x+\frac{2}{3}y \right)^2=\left(\frac{1}{2}x\right)^2+2\times\frac{1}{2}x\times\frac{2}{3}y+\left(\frac{2}{3}y\right)^2=\frac{1}{4}x^2+\frac{2}{3}xy+\frac{4}{9}y^2.}

\latex{a}

\latex{b}

\latex{a}

\latex{b}

\latex{a^2}

\latex{a\times b}

\latex{b^2}

\latex{(a+b)^2=a^2+2ab+b^2}

Figure 3

Square of a difference

\latex{(a-b)^2}=\latex{(a-b)(a-b)}=\latex{a^2-ab-ba+b^2}=\latex{a^2-2ab+b^2}

\latex{(a-b)^2 = a^2-2ab+b^2}

E.g.

\latex{(y-4)^2=y^2-2\times y\times4+4^2=y^2-8y+16;\\[10pt]}

\latex{(3a-x)^2=(3a)^2-2\times3a\times x+x^2=9a^2-6ax+x^2;\\[10pt]}
\latex{(7x-5y)^2=(7x)^2-2\times7x \times5y+(5y)^2=49x^2-70xy+25y^2;\\[10pt]}

\latex{\left( \frac{2}{5}a-\frac{1}{4}b\right)^2=\left(\frac{2}{5}a\right)^2-2\times\frac{2}{5}a\times\frac{1}{4}b+\left(\frac{1}{4}b\right)^2=\frac{4}{25}a^2-\frac{1}{5}ab+\frac{1}{16}b^2.}

\latex{a}

\latex{b}

\latex{a\times b}

\latex{(a-b)^2}

\latex{a\times b}

\latex{b^2}

\latex{(a-b)^2=a^2-2ab+b^2}

Figure 4

Square of a trinimial sum

\latex{(a+b+c)^2}=\latex{[(a+b)+c]^2=}\latex{(a+b)^2+2(a+b)c+c^2=}

\latex{=a^2+2ab+b^2+2ac+2bc+c^2}\latex{=a^2+b^2+c^2+2ab+2ac+2bc.}

\latex{(a+b+c)^2 = a^2+b^2+c^2+2ab+2ac+2bc}

According to this the square of a trinomial sum equals to the sum of the following: the square of the terms and twice the products of the terms in pairs.

E.g.

\latex{(3x+2y+z^2)^2}=\latex{(3x)^2+(2y)^2+(z^2)^2+2\times3x\times2y+2\times 3x\times z^2+}\latex{+2\times2y\times z^2=9x^2+4y^2+z^4+12xy+6xz^2+4yz^2;\\[10pt]}

\latex{(4a-3b+2c)^2=(4a+(-3b)+2c)^2=}\latex{=(4a)^2+(-3b)^2+(2c)^2+2\times4a\times(-3b)+2\times4a\times2c+2\times(-3b)\times2c=} \latex{=16a^2+9b^2+4c^2-24ab+16ac-12bc}.

The previous result can be used not only for the square of a trinomial sum. Squaring an arbitrary polynomial gives a similar form: each term should be squared and twice the products of the terms paired in all possible ways should be added to the squares.

Third power (or cube) of a binomial sum

\latex{(a+b)^3}\latex{=(a+b)(a+b)^2=} \latex{(a+b)(a^2+2ab+b^2)=}

\latex{=a^3+2a^2b+ab^2+ba^2+2ab^2+b^3 =} \latex{a^3+3a^2b+3ab^2+b^3}

\latex{(a+b)^3=a^3+3a^2b+3ab^2+b^3}

Our result can also be written in the form of

\latex{(a+b)^3=a^3+b^3+3ab(a+b)},

which can be often used when solving certain exercises.

E.g.

\latex{(x+2)^3=} \latex{x^3+3\times x^2\times 2+3\times x\times2^2+2^3=x^3+6x^2+12x+8;\\[10pt]}

\latex{(2a+3b)^3=(2a)^3+3\times(2a)^2\times 3b+3\times 2a\times (3b)^2+(3b)^3=}\latex{=8a^3+36a^2b+54ab^2+27b^3;}
\latex{[a+(-b)]^3=a^3+3a^2(-b)+3a(-b)^2+b^3=} \latex{a^3-3a^2b+3ab^2-b^3}.

\latex{a}

\latex{b}

\latex{(a+b)^3=}
\latex{=a^3+3a^2b+3ab^2+b^3}

Figure 5

Third power (or cube) of a binomial difference

\latex{(a-b)^3}=\latex{(a-b)(a-b)^2}=\latex{(a-b)(a^2-2ab+b^2)=}
\latex{=a^3-2a^2b+ab^2-ba^2+2ab^2-b^3}= \latex{a^3-3a^2b+3ab^2-b^3}

\latex{(a-b)^3=a^3-3a^2b+3ab^2-b^3}

Even this identity can be written in the form of

\latex{(a-b)^3=a^3-b^3-3ab(a-b)}.

E.g.
\latex{(3a-1)^3=} \latex{(3a)^3-3\times(3a)^2\times 1+3\times(3a)\times1^2-1^3=} \latex{27a^3-27a^2+9a-1;}

\latex{\left( \frac{1}{4}x-\frac{2}{3}y\right)^3=\left(\frac{1}{4}x\right)^3-3\times\left(\frac{1}{4}x\right)^2\times\frac{2}{3}y+3\times\frac{1}{4}x\times\left(\frac{2}{3}y\right)^2-\left(\frac{2}{3}y\right)^3=\\[10pt]} \latex{=\frac{1}{64}x^3-\frac{1}{8}x^2y+\frac{1}{3}xy^2-\frac{8}{27}y^3;}

\latex{(-a-b)^3=(-a)^3-3\times(-a)^2b+3\times(-a)b^2-b^3} \latex{=-a^3-3a^2b-3ab^2-b^3=-(a+b)^3.}

The product of the sum and of the difference of two terms

\latex{(a+b)(a-b)} \latex{=a^2-ab+ba-b^2=} \latex{a^2-b^2}

\latex{(a+b)\times(a-b)=a^2-b^2}

The product of a binomial sum and of the difference of the same terms equals to the difference of the squares of the first and the second term.

E.g.

\latex{(a+3)(a-3)=} \latex{a^2-9;\\[10pt]}
\latex{(2x-1)(2x+1)=} \latex{4x^2-1;\\[10pt]}
\latex{(5a-6b)(5a+6b)=} \latex{25a^2-36b^2}.

\latex{a}

\latex{b}

\latex{a-}\latex{b}

\latex{b}

\latex{a}

\latex{b^2}

\latex{(a+b)\times(a-b)=a^2-b^2}

Figure 6

Calculation tricks

\latex{79^2=(80-1)^2=80^2-2\times80\times1+1^2=6,400-160+1=6,241;}
\latex{61^2=(60+1)^2=60^2+2\times60\times1+1^2=3,600+120+1=3,721;}
\latex{11^3= (10+1)^3=10^3+3\times10^2\times1+3\times10\times1^2+1^3=1,331;}
\latex{103\times97=(100+3)(100-3)= 100^2-3^2=10,000-9=9,991.}

Two more special identities

\latex{(a-b)(a^2+ab+b^2)=a^3+a^2b+ab^2-a^2b-ab^2-b^3=a^3-b^3}, i.e.

\latex{(a-b)(a^2+ab+b^2)=a^3-b^3.}

\latex{(a+b)(a^2-ab+b^2)=a^3-a^2b+ab^2+a^2b-ab^2+b^3=a^3+b^3}, i.e.
\latex{(a+b)(a^2-ab+b^2)=a^3+b^3.}

Example 1

The square of which expression is the following sum:

\latex{ 4a^2+4a+1?}

Solution

\latex{4a^2+4a+1=(2a+1)^2}

Note: Obviously \latex{4a^2+4a+1=(-2a-1)^2} is also acceptable, since \latex{(-a-b)^2=[-(a+b)]^2=(a+b)^2.}

Example 2

The square or cube of which expressions are the below sums?

\latex{9a^2-6a+1;} \latex{16x^4+8x^2+1;} \latex{x^3+3x^2+3x+1}

Solution

\latex{9a^2-6a+1=(3a-1)^2=(1-3a)^2;\\[10pt]}
\latex{16x^4+8x^2+1=(4x^2+1)^2;\\[10pt]}
\latex{x^3+3x^2+3x+1=(x+1)^3.}

Example 3

Let us carry out the following operations.

\latex{(3a-b)^2-(3a+2b)(3a-2b)+(a+2b)^2-(a-b)^2}

Solution

\latex{(3a-b)^2-(3a+2b)(3a-2b)+(a+2b)^2-(a-b)^2=} \latex{=9a^2-6ab+b^2-(9a^2-4b^2)+a^2+4ab+4b^2-(a^2-2ab+b^2)=}\latex{=9a^2-6ab+b^2-9a^2+4b^2+a^2+4ab+4b^2-a^2+2ab-b^2=8b^2}.

Example 4

Let us transform the following quadratic expressions so that the variable appears only in the square of a binomial. (Completing the square.)

\latex{x^2+6x+13}

\latex{x^2-10x+13}

\latex{x^2+8x+2}

\latex{x^2-13x+7}

\latex{2x^2-20x+54}

Solution

\latex{x^2+6x+13=(x^2+2\times x\times 3+3^2)-3^2+13=(x+3)^2+4;\\[10pt]}
\latex{x^2-10x+13=(x^2-2\times x\times 5+5^2)-5^2+13=(x-5)^2-12;\\[10pt]}
\latex{x^2+8x+2=(x^2+8x+16)-16+2=(x+4)^2-14;\\[10pt]}
\latex{x^2-13x+7=\left(x^2-13x+\left(\frac{13}{2}\right)^2\right)-\left(\frac{13}{2}\right)^2+7=(x-6.5)^2-35.25;\\[10pt]}
\latex{2x^2-20x+54=2(x^2-10x+27)=}\latex{=2[(x^2-10x+25)-25+27]=2[(x-5)^2+2]=2(x-5)^2+4.}

The transformation applied in the example is often used when rewriting quadratic expressions; this process is called completing the square.

Example 5

The product of two numbers is \latex{ 23 }, their sum is \latex{ 12 }. Give the sum of the squares of the two numbers.

Solution

Let the numbers be a and b, then

\latex{a\times b=23} and \latex{a+b=12.}

We are looking for \latex{a^2+b^2.}
Since

\latex{(a+b)^2=a^2+2ab+b^2,}

then

\latex{a^2+b^2=(a+b)^2-2ab=12^2-2\times23=144-46=98.}

Example 6

The product of two numbers is \latex{ 12 }, their sum is \latex{ 23 }. Calculate the sum of the cubes of the two numbers.

Solution

Let the numbers be a and b, then

\latex{a\times b=12} and \latex{a+b=23.}

We are looking for \latex{a^3+b^3.}
Since

\latex{(a+b)^3=a^3+b^3+3ab(a+b),}

then

\latex{a^3+b^3=(a+b)^3-3ab(a+b)=23^3-3\times 12\times 23=12,167-828=11,339.}

Exercises

{{exercise_number}}. Perform the following expressions.

a) \latex{(6a-5b)^2}

b) \latex{(10a+2b)^2}

c) \latex{(8x+3y)^2}

d) \latex{(7x^2+3)^2}

e) \latex{(a-9b^3)^2}

f) \latex{(4a^2-5b^5)^2}

g) \latex{\left(\frac{5}{7}a+\frac{1}{3}b\right)^2}

h) \latex{\left(\frac{7}{11}x^4-\frac{3}{8}y^3\right)^2}

{{exercise_number}}. Perform the following expressions.

a) \latex{(2a+4b+c^3)^2}

b) \latex{(5x-3y^2+2)^2}

c) \latex{\left(6x-\frac{2}{3}y-4z^2\right)^2}

d) \latex{\left(\frac{3}{4}a-\frac{2}{3}b+\frac{1}{7}\right)^2}

e) \latex{(2a-3b+4c-d)^2}

{{exercise_number}}. Perform the following expressions.

a) \latex{(3x+y)^3}

b) \latex{(4a^2-2b)^3}

c) \latex{\left(\frac{1}{2}x+2y\right)^3}

d) \latex{\left(\frac{2}{5}x-\frac{1}{3}y^3\right)^3}

e) \latex{\left(\frac{1}{4}a^2+5b\right)^3}

f) \latex{\left(\frac{3}{5}a-0.4b\right)^3}

{{exercise_number}}. Perform the following expressions.

a) \latex{(7x-6y)(7x+6y)}

b) \latex{(3a+5b)(3a-5b)}

c) \latex{\left(\frac{1}{5}x-7\right)\times\left(\frac{1}{5}x+7\right)}

d) \latex{(x^2-6a)(x^2+6a)}

e) \latex{(1.2a^3-9b^2)(1.2a^3+9b^2)}

f) \latex{(8x^2y-3xy^2)(8x^2y+3xy^2)}

g) \latex{\left(\frac{1}{2}a+\frac{11}{2}b\right)\times(0.5a-5.5b)}

h) \latex{\left(1.2x^2-\frac{2}{5}y^2\right)\times\left(\frac{6}{5}x^2+0.4y^2\right)}

{{exercise_number}}. Perform the following expressions.

\latex{(2x+y)^3-(6x+y)(6x-y)+(x^2-5)^2}
\latex{(4x-y)(4x+y)+(2x-y)^2-(5x+2y)(5x-2y)}
\latex{(5a+b)^3-(5a-b)^3-5(4a-3b)(4a+3b)}

\latex{\left(\frac{1}{2}x-1\right)^3+\left(2+\frac{3}{2}x\right)^3-\left(\frac{5}{2}x+3\right)\times \left(\frac{5}{2}x-3\right)}
\latex{\left(\frac{2}{3}a+3\right)^2-\left(\frac{2}{3}a-1\right)^2+\left(\frac{5}{3}a-4\right)\times \left(\frac{5}{3}a+4\right)}

{{exercise_number}}. Complete the squares in the below expressions.

\latex{x^2-6x+10}

\latex{x^2+12x+39}

\latex{x^2-7x+13}

\latex{x^2+21x+21}

\latex{3x^2-6x+8}

\latex{-2x^2+6x-1}

{{exercise_number}}. Perform the following expressions.

\latex{(a-2)(a^2+2a+4)}

\latex{(b+3)(b^2-3b+9)}

\latex{(2x-3)(4x^2+6x+9)}

{{exercise_number}}. Calculate cleverly.

\latex{\frac{437^2-363^2}{537^2-463^2}}

\latex{96\times 104}

{{exercise_number}}. Solve the below exercises.

Give the prime factorisation of \latex{899}.
Show that \latex{7,778^2-2,223^2=55,555,555.}

Quiz

Which number is greater: \latex{632,757 × 632,763} or \latex{632,760^2?}

Mathematics 9.

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