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Squared binomial

A binomial is one that has two terms, which are separated by signs. We can find them separated by parentheses and it will indicate that everything inside the parentheses will be squared. The rule tells us that the result of the squared binomial will be the first term squared plus twice the first term by the second term plus the second term squared

$(a+b)^2=a^2+2ab+b^2$

These rules will always be fulfilled and it is not necessary to do the multiplications, making it easier to solve the problem

• The middle sign of our binomial squared will be the first sign of our result. In the result, the signs of the first and third terms will always be positive

$(a+b)^2=a^2+2ab+b^2$ $(a-b)^2=a^2-2ab+b^2$ $(-a+b)^2=-(a-b)^2=(a-b)^2=a^2-2ab+b^2$ $(-a-b)^2=-(a+b)^2=(a+b)^2=a^2+2ab+b^2$ $(a+b)(a-b)=a^2+ab-ab-b^2=a^2-b^2$ $(x+a)(x+b)=x^2+xa+xb+ab=x^2+(a+b)x+ab$ $(ax+b)(cx+d)=acx^2+adx+bcx+bd=acx^2+(ad+bc)x+bd$

• When you are squaring a fraction, both the numerator and denominator will be squared independently

$\left(\frac{a}{b}\right)^2=\frac{a^2}{b^2}$

Material: Binomial squared

Binomial squared

Squaring a+b is equivalent to multiplying this binomial by itself and we will have:

$(a+b)^2=a^2+2ab+b^2$

Examples:

1. $(y+6)^2=y^2+12y+36$
2. $(1+3z)^2=1+6z+9z^2=9z^2+6z+1$
3. $(4xy^3+5y)^2=16x^2y^6+40xy^4+25y^2$
4. $(mn^2+n)^2=m^2n^4+2mn^3+n^2$
5. $(a^2b+b)^2=a^4b^2+2a^2b^2+b^2$

Binomial to the nth power

When we raise a factor (such as a binomial with two terms) to the square, it means that we are going to multiply that factor twice, when cubed it would multiply itself 3 times and so on. To make these multiplications easier we've something called Pascal's triangle that will give us the coefficients that we're going to have in our binomials raised to the x power

Pascal's triangle is made by starting the first line (point) with a 1, the second line with two terms (both ones), the third with a one in each corner and in the middle the sum of the terms above and so on, each line

• The exponent to which our expression is raised will give us the maximum value of the degree of our variables (which is reflected in the terms of the corners)

• Whenever we have a negative sign in our binomial, the signs will alternate (positive first) in the result

Pascal's triangle	Power of a addition
1	$(a+b)^0=1$
1 1	$(a+b)^1= 1a+1b$
1 2 1	$(a+b)^2= 1a^2+2ab+1b^2$
1 3 3 1	$(a+b)^3= 1a^3+3a^2b+3ab^2+1b^3$
1 4 6 4 1	$(a+b)^4= 1a^4+4a^3b+6a^2b^2+4ab^3+1b^4$
1 5 10 10 5 1	$(a+b)^5= 1a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+1b^5$
...	...

Pascal's triangle	Power of a subtraction
1	$(a-b)^0=1$
1 1	$(a-b)^1= 1a-1b$
1 2 1	$(a-b)^2= 1a^2-2ab+1b^2$
1 3 3 1	$(a-b)^3= 1a^3-3a^2b+3ab^2-1b^3$
1 4 6 4 1	$(a-b)^4= 1a^4-4a^3b+6a^2b^2-4ab^3+1b^4$
1 5 10 10 5 1	$(a-b)^5= 1a^5-5a^4b+10a^3b^2-10a^2b^3+5ab^4-1b^5$
...	...

Conjugate binomials

A binomial is a product between two terms and conjugate means that we will have the same terms of the binomial on one side as on the other with opposite signs (one positive and one negative)

The rule tells us that by doing the product between these terms we will have a difference of perfect squares where we square our first term minus the second term squared

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

Material: Binomials to the nth power and conjugates

Cube Binomial

Cubing a+b is equivalent to multiplying this binomial by itself three times and we will have:

$(a+b)^3=(a+b)(a+b)(a+b)=a^3+3a^2b+3ab^2+b^3$

Examples

1. $(x+2)^3=x^3+6x^2+12x+8$
2. $(m+3)^3=m^3+9m^2+27m+9$
3. $(3x+1)^3=27x^3+27x^2+9x+1$
4. $(1-b^2)^3=1-3b^2+3b^4-b^6$
5. $(2x+3y)^3=8x^3+36x^2y+54xy^2+27y^3$

Difference of squares

Be the quotient

$\frac{a^2-b^2}{a+b}=\frac{(a-b)\cancel{(a+b)}}{\cancel{(a+b)}}=(a-b)$

we've

$R=a-b$

Be the quotient

$\frac{a^2-b^2}{a-b}=\frac{\cancel{(a-b)}(a+b)}{\cancel{(a-b)}}=(a+b)$

we've

$R=a+b$

Rule:

• The difference of the squares of two quantities divided by the sum of the quantities equals the difference of the quantities

• The difference of the squares of two quantities divided by the difference of the quantities equals the sum of the quantities

Examples:

1. $\frac{x^2-2}{x+\sqrt{2}}=x-\sqrt{2}$
2. $\frac{y^2-4x^2}{y+2x}=y-2x$
3. $\frac{4a^2-9b^2c^4}{2a+3bc^2}=2a-3bc^2$
4. $\frac{x^4y^6-4a^8b^{10}}{x^2y^3+2a^4b^5}=x^2y^3-2a^4b^5$
5. $\frac{x^{2n}-y^{2n}}{x^n+y^n}=x^n-y^n$
6. $\frac{1-(x+y)^2}{1+(x+y)}=1-(x+y)$
7. $\frac{(b+x))^2-9}{(b+x)+3}=(b+x)-3$

Difference of the cubes

Be the quotient

$$\begin{split} \frac{a^3-b^3}{a+b} & =\frac{\cancel{(a+b)}(a^2-ab+b^2)}{\cancel{(a+b)}}\\ & =a^2-ab+b^2 \end{split}$$

we've

$R=a^2-ab+b^2$

Be the quotient

$$\begin{split} \frac{a^3-b^3}{a-b} & =\frac{\cancel{(a-b)}(a^2+ab+b^2)}{\cancel{(a-b)}}\\ & =a^2+ab+b^2 \end{split}$$

we've

$R=a^2+ab+b^2$

Rule:

• The sum of the cubes of two quantities divided by the sum of the quantities is equal to the square of the first quantity, minus the product of the first and the second, plus the square of the second quantity

• The difference of the cubes of two quantities divided by the difference of the quantities is equal to the square of the first quantity, plus the product of the first and the second, plus the square of the second quantity

Examples:

1. $\frac{1+y^3}{1+y}=1-y+y^2=y^2-y+1$
2. $\frac{8y^3-1}{2y-1}=4y^2+2y+1$
3. $\frac{27z^3-125y^3}{3z-5y}=9z^2+15yz+25y^2$
4. $\frac{27a^6+1}{3a^2+1}=9a^4-3a^2+1$
5. $\frac{b^6+1}{b^2+1}=b^4+b^2+1$