Question 196043
# 1




{{{(2x-3y)^2}}} Start with the given expression.



{{{(2x-3y)(2x-3y)}}} Expand. Remember something like {{{x^2=x*x}}}.



Now let's FOIL the expression.



Remember, when you FOIL an expression, you follow this procedure:



{{{(highlight(2x)-3y)(highlight(2x)-3y)}}} Multiply the <font color="red">F</font>irst terms:{{{(2*x)*(2*x)=4*x^2}}}.



{{{(highlight(2x)-3y)(2x+highlight(-3y))}}} Multiply the <font color="red">O</font>uter terms:{{{(2*x)*(-3*y)=-6*x*y}}}.



{{{(2x+highlight(-3y))(highlight(2x)-3y)}}} Multiply the <font color="red">I</font>nner terms:{{{(-3*y)*(2*x)=-6*x*y}}}.



{{{(2x+highlight(-3y))(2x+highlight(-3y))}}} Multiply the <font color="red">L</font>ast terms:{{{(-3*y)*(-3*y)=9*y^2}}}.



{{{4*x^2-6*x*y-6*x*y+9*y^2}}} Now collect every term to make a single expression.



{{{4*x^2-12*x*y+9*y^2}}} Now combine like terms.



So {{{(2x-3y)^2}}} FOILs to {{{4*x^2-12*x*y+9*y^2}}}.



In other words, {{{(2x-3y)^2=4*x^2-12*x*y+9*y^2}}}.



<hr>


# 2




{{{(6x-7)(6x+7)}}} Start with the given expression.



Now let's FOIL the expression.



Remember, when you FOIL an expression, you follow this procedure:



{{{(highlight(6x)-7)(highlight(6x)+7)}}} Multiply the <font color="red">F</font>irst terms:{{{(6*x)*(6*x)=36*x^2}}}.



{{{(highlight(6x)-7)(6x+highlight(7))}}} Multiply the <font color="red">O</font>uter terms:{{{(6*x)*(7)=42*x}}}.



{{{(6x+highlight(-7))(highlight(6x)+7)}}} Multiply the <font color="red">I</font>nner terms:{{{(-7)*(6*x)=-42*x}}}.



{{{(6x+highlight(-7))(6x+highlight(7))}}} Multiply the <font color="red">L</font>ast terms:{{{(-7)*(7)=-49}}}.



{{{36*x^2+42*x-42*x-49}}} Now collect every term to make a single expression.



{{{36*x^2-49}}} Now combine like terms.



So {{{(6x-7)(6x+7)}}} FOILs to {{{36*x^2-49}}}.



In other words, {{{(6x-7)(6x+7)=36*x^2-49}}}.



Note: this is a difference of squares.


<hr>


# 3


Let's simplify this expression using synthetic division



Start with the given expression {{{(3x^3 - x^2 + 10x - 4)/(x+3)}}}


First lets find our test zero:


{{{x+3=0}}} Set the denominator {{{x+3}}} equal to zero


{{{x=-3}}} Solve for x.


so our test zero is -3



Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.<TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>3</TD><TD>-1</TD><TD>10</TD><TD>-4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 3)

<TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>3</TD><TD>-1</TD><TD>10</TD><TD>-4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD></TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -3 by 3 and place the product (which is -9)  right underneath the second  coefficient (which is -1)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>3</TD><TD>-1</TD><TD>10</TD><TD>-4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-9</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD></TD><TD></TD><TD></TD></TR></TABLE>

    Add -9 and -1 to get -10. Place the sum right underneath -9.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>3</TD><TD>-1</TD><TD>10</TD><TD>-4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-9</TD><TD></TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>-10</TD><TD></TD><TD></TD></TR></TABLE>

    Multiply -3 by -10 and place the product (which is 30)  right underneath the third  coefficient (which is 10)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>3</TD><TD>-1</TD><TD>10</TD><TD>-4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-9</TD><TD>30</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>-10</TD><TD></TD><TD></TD></TR></TABLE>

    Add 30 and 10 to get 40. Place the sum right underneath 30.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>3</TD><TD>-1</TD><TD>10</TD><TD>-4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-9</TD><TD>30</TD><TD></TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>-10</TD><TD>40</TD><TD></TD></TR></TABLE>

    Multiply -3 by 40 and place the product (which is -120)  right underneath the fourth  coefficient (which is -4)

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>3</TD><TD>-1</TD><TD>10</TD><TD>-4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-9</TD><TD>30</TD><TD>-120</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>-10</TD><TD>40</TD><TD></TD></TR></TABLE>

    Add -120 and -4 to get -124. Place the sum right underneath -120.

    <TABLE cellpadding=10><TR><TD>-3</TD><TD>|</TD><TD>3</TD><TD>-1</TD><TD>10</TD><TD>-4</TD></TR><TR><TD></TD><TD>|</TD><TD></TD><TD>-9</TD><TD>30</TD><TD>-120</TD><TD></TD></TR><TR><TD></TD><TD></TD><TD>3</TD><TD>-10</TD><TD>40</TD><TD>-124</TD></TR></TABLE>

Since the last column adds to -124, we have a remainder of -124. This means {{{x+3}}} is <b>not</b> a factor of  {{{3x^3 - x^2 + 10x - 4}}}

Now lets look at the bottom row of coefficients:


The first 3 coefficients (3,-10,40) form the quotient


{{{3x^2 - 10x + 40}}}


and the last coefficient -124, is the remainder, which is placed over {{{x+3}}} like this


{{{-124/(x+3)}}}




Putting this altogether, we get:


{{{3x^2 - 10x + 40-124/(x+3)}}}


So {{{(3x^3 - x^2 + 10x - 4)/(x+3)=3x^2 - 10x + 40-124/(x+3)}}}


which looks like this in remainder form:


{{{(3x^3 - x^2 + 10x - 4)/(x+3)=3x^2 - 10x + 40}}} remainder -124



You can use this <a href=http://calc101.com/webMathematica/long-divide.jsp>online polynomial division calculator</a> to check your work