FOILing and Explanation for FOIL

FOILing and Explanation for FOIL exercise

• Calculate $(25 + 2x)(20 + x)$ using the FOIL method.

  Hints

  FOIL is an easy way to remember how to multiply two binomials:

  F first multiply the first terms.

  O next multiply the outer terms.

  I now the inner terms.

  L last multiply the last terms.

  $25$ and $20$ are the first terms of the two binomials.

  Here you see an example using the FOIL method.

  Solution

  FOIL is an easy way to remember how to multiply two binomials.

  F stands for first, meaning multiply the first terms $25\times 20=500$.

  O stands for outer, meaning multiply the outer terms $25\times x=25x$.

  I stands for inner, meaning multiply the inner terms $2x\times 20=40x$.

  L stands for last, meaning multiply the last terms $2x\times x=2x^2$.

  But the calculation isn't finished yet! We still have to combine the like terms and arrange the terms such that the resulting polynomial starts with the highest degree term and ends with the lowest one:

  $(25+2x)(20+x)=500+25x+40x+2x^2=500+65x+2x^2=2x^2+65x+500$.

• Determine the area of the zoo via the distributive property and the FOIL method.

  Hints

  The FOIL method only works for the multiplication of two binomials.

  The distributive property is $a(b\pm c)=ab\pm ac$.

  Remember to combine like terms. For example, $3x+5x=8x$.

  Solution

  The total area of the fields on the old map can be represented as the product $(25+2x)(20+x+3x)$.

  The second term is a trinomial, not a binomial, so the FOIL method cannot be applied. But we can use the distributive property, $a(b\pm c)=ab\pm ac$, to get

  $(25+2x)(30+x+3x)=(25+2x)(20)+(25+2x)(x)+(25+2x)(3x)$.

  We use the distributive property once again to get

  $(25+2x)(20)+(25+2x)(x)+(25+2x)(3x)=500+40x+25x+2x^2+75x+6x^2$.

  Next we combine like terms:

  $500+40x+25x+2x^2+75x+6x^2=500+140x+8x^2$,

  and rearrange the powers according to the degree,

  $500+140x+8x^2=8x^2+140x+500$.

  We could have also combined like terms in $20+x+3x$ to get $20+4x$. Then we are multiplying two binomials, $(25+2x)(20+4x)$, and we can use the FOIL method:

  1. F multiply the first terms $25\times 20=500$.
  2. O multiply the outer terms $25\times 4x=100x$.
  3. I multiply the inner terms $2x\times 20=40x$.
  4. L multiply the last terms $2x\times 4x=8x^2$.

  To finish the the calculation we still have to combine the like terms and arrange the terms so that the resulting term starts with the highest degree and ends with the lowest one:

  $(25+2x)(20+4x)=500+100x+40x+8x^2=500+140x+8x^2=8x^2+140x+500$.

  We can now see that with either method, we get the same result in the end. And John now knows the total area of the fields on the old map and can start planning the construction of his dream zoo!

• Identify the right formula for calculating the area.

  Hints

  First establish the multiplication.

  For example the leftmost one is $(30+3x)(45+x)$.

  Next combine like terms if there are any.

  You can use the FOIL method for all these examples.

  F stands for multiply the first terms.

  O stands for multiply the outer terms.

  I stands for multiply the inner terms.

  L stands for multiply the last terms.

  Solution

  For calculating the regarding areas we use the FOIL method

  F multiply the first terms.

  O multiply the outer terms.

  I multiply the inner terms.

  L multiply the last terms.

  $~$

  For the leftmost one we have:

  $(30+3x)(45+x)$

  1. F $30\times 45=1350$
  2. O $30\times x=30x$
  3. I $3x\times 45=135x$
  4. L $3x\times x=3x^2$

  Next we add these terms, combine like terms, and rearrange the terms by degree:

  $(30+3x)(45+x)=1350+30x+135x+3x^2=1350+165x+3x^2=3x^2+165x+1350$.

  $~$

  For the next example we first simplify the second factor, and then use the FOIL method:

  $(30+3x)(45+x+2x)=(30+3x)(45+3x)$

  1. F $30\times 45=1350$
  2. O $30\times 3x=90x$
  3. I $3x\times 45=135x$
  4. L $3x\times 3x=9x^2$

  Next we add these terms, combine the like terms, and rearrange the terms by degree:

  $(30+3x)(45+3x)=1350+90x+135x+9x^2=1350+225x+9x^2=9x^2+225x+1350$.

  $~$

  For the next, we have:

  $(40+2x)(15+4x)$

  1. F $40\times 15=600$
  2. O $40\times 4x=160x$
  3. I $2x\times 15=30x$
  4. L $2x\times 4x=8x^2$

  Next we add these terms, combine the like terms, and rearrange the terms by degree:

  $(40+2x)(15+4x)=600+160x+30x+8x^2=600+190x+8x^2=8x^2+190x+600$.

  $~$

  For the rightmost one, we first have to simplify one factor before using the FOIL method:

  $(40+2x)(15+4x+10+2x)=(40+2x)(25+6x)$

  1. F $40\times 25=1000$
  2. O $40\times 6x=240x$
  3. I $2x\times 25=50x$
  4. L $2x\times 6x=12x^2$

  Next we add these terms, combine the like terms, and rearrange the terms by degree:

  $(40+2x)(15+4x)=1000+240x+50x+12x^2=1000+290x+12x^2=12x^2+290x+1000$.

• Calculate the new area of the zoo.

  Hints

  Combine the like terms in order to use the FOIL method.

  You have to calculate the product $(25+2x)(20+6x)$.

  Here is an example using the FOIL method, combining like terms, and rearranging the powers.

  Solution

  To calculate the total needed area for John's Zoo we have to multiply the length $25+2x$ with the width $20+x+3x+2x$.

  To use the FOIL method we first have to simplify the width term to $20+6x$.

  Now we have to calculate $(25+2x)(20+6x)$:

  1. F $25\times 20=500$
  2. O $25\times 6x=150x$
  3. I $2x\times 20=40x$
  4. L $2x\times 6x=12x^2$

  The calculation isn't finished. Next we add these terms, combine the like terms and rearrange the terms depending on the degrees:

  $(25+2x)(20+6x)=500+150x+40x+12x^2=500+190x+12x^2=12x^2+190x+500$.

• Decide when the FOIL method can be used.

  Hints

  F stands for multiply the first terms.

  O stands for multiply the outer terms

  I stands for multiply the inner terms.

  L stands for multiply the last terms.

  A binomial is the result of adding or subtracting two monomials. For example:

  • $2x$ and $3$ are monomials.
  • $2x+3$ is a binomial.

  A trinomial is the result of adding or subtracting three monomials.

  Solution

  FOIL is an easy way to remember how to multiply two binomials.

  And it only works when you are multiplying two binomials.

  F stands for multiply the first terms.

  O stands for multiply the outer terms

  I stands for multiply the inner terms.

  L stands for multiply the last terms.

• Find the mistakes in the calculations.

  Hints

  You can use the FOIL method also for differences.

  Pay attention to the sign.

  Consider this example:

  $\begin{array}{rcl} (1+x)(3-x)&=&~1\times 3+1\times (-x)+x\times 3+x\times (-x)\\ &=&~3-x+3x-x^2 \end{array}$

  If necessary, first simplify the terms.

  There are ten mistakes in total.

  Solution

  Remember the following:

  • For using the FOIL method we have to simplify the factors if necessary.
  • You can also use the FOIL method for differences. But you have to pay attention to the sign.

  $\begin{array}{rcl} (12+2x)(10-4x)&=&~12\times 10+12\times (-4x)+2x\times 10+2x\times (-4x)\\ &=&~120-48x+20x-8x^2\\ &=&~120-28x-8x^2\\ &=&~-8x^2-28x+120 \end{array}$

  $\begin{array}{rcl} (15-x)(20+3x)&=&~15\times 20+15\times 3x+(-x)\times 20+(-x)\times 3x\\ &=&~300+45x-20x-3x^2\\ &=&~300+25x-3x^2\\ &=&~-3x^2+25x+300 \end{array}$

  $\begin{array}{rcl} (25-2x)(40-2x)&=&~25\times 40+25\times (-2x)+(-2x)\times 40+(-2x)\times (-2x)\\ &=&~1000-50x-80x+4x^2\\ &=&~1000-130x+4x^2\\ &=&~4x^2-130x+1000 \end{array}$

  Lastly we still have a multiplication where we first have to simplify both factors: $(25+2x-4x)(20+3x-10-7x)=(25-2x)(10-4x)$.

  $\begin{array}{rcl} (25-2x)(10-4x)&=&~25\times 10+25\times (-4x)+(-2x)\times 10+(-2x)\times (-4x)\\ &=&~250-100x-20x+8x^2\\ &=&~250-120x+8x^2\\ &=&~8x^2-120x+250 \end{array}$