Multiplying Special Case Polynomials

Multiplying Special Case Polynomials exercise

• Calculate the term ${(a+b)^2}$.

  Hints

  Imagine you cut the square with side lengths ${a+b}$ into two squares and two rectangles.

  The area of the original square is the same as the sum of the two smaller squares and two rectangles.

  You can also use the FOIL method to simplify ${(a+b)(a+b)}$.

  Solution

  The expression ${(a+b)(a+b)}$, or ${ (a + b)^2}$, can be simplified using the area model.

  Imagine a square with side lengths ${a+b}$; its area is ${(a+b)^2}$.

  You can divide this square into two smaller squares with sides $ a$ and $b$, respectively, as well as two rectangles with a height $a$ and length $b$.

  The areas of the squares are ${a^2}$ and ${b^2}$, while each rectangle has an area equal to ${ ab}$.

  Combining the like terms gives us ${(a+b)^2=a^2+2ab+b^2}$.

• Summarize the multiplication of special binomials.

  Hints

  The difference of two squares is ${(a+b)(a-b)}$.

  You can simplify the expansion of binomials by using the FOIL method.

  For ${ (a+b)(a+b)}$, use the FOIL method:

  F multiply the first ${a \times a}$

  O multiply the outer ${a \times b}$

  I multiply the inner ${b \times a}$

  L multiply the last ${b \times b}$

  Solution

  How can you remember the pattern for multiplying of binomials?

  Perfect Square Binomials

  ${ \begin{array}{rcl} (a + b)(a + b) &=&\\ &=& a^2 + ab + ab + b^2\\ &=& a^2 + 2ab + b^2 \end{array}}$

  $~$

  ${\begin{array}{rcl} (a - b)(a - b) &=&\\ &=& a^2 - ab - ab + b^2\\ &=& a^2 - 2ab + b^2 \end{array}}$

  Difference of Two Squares

  ${\begin{array}{rcl} (a + b)(a - b) &=&\\ &=& a^2 - ab + ab + b^2\\ &=& a^2-b^2 \end{array}}$

• Use the area model to simplify the expression.

  Hints

  ${ \begin{array}{rcl} (4y)^2 &=&\\ 4^2 \times y^2 &=& 16y^2 \end{array}}$

  Draw the square and write the areas in the corresponding squares or rectangles.

  Solution

  If you have a perfect square binomial such as $(x + a)^2$, you can approach this in two ways:

  1.$~$Using FOIL

  ${\begin{array}{rcl} (x + a)^2 &=&\\ &=& (x + a)(x + a)\\ &=& x^2 + ax + ax + a^2\\ &=& x^2 + 2ax + a^2\\ ~\\ \end{array}}$

  2.$~$Using the FOIL shortcut for Perfect Square Binomials

  ${ \begin{array}{rcl} ~\\ (x + a)^2 &=&\\ &=& x^2 + 2(ax) + a^2\\ \end{array}}$

  $~$

  ${ \begin{array}{rcl} (x + 5)^2&=&\\ &=& x^2 + 2 \times 5x + 5^2\\ &=& x^2 + 10x + 25 \end{array}}$

  $~$

  ${ \begin{array}{rcl} (3 + 3x)^2&=&\\ &=&3^2 + 2 \times 3 \times 3x + (3x)^2\\ &=& 9x^2 + 18x + 9 \end{array}}$

  $~$

  ${ \begin{array}{rcl} (1 + 4y)^2&=&\\ &=&1^2 + 2 \times 1 \times 4y + (4y)^2\\ &=& 16y^2 + 8y + 1 \end{array}}$

  $~$

  ${ \begin{array}{rcl} (2x + 4)^2&=&\\ &=&(2x)^2 + 2 \times 2x \times 4 + 4^2\\ &=& 4x^2+16x+16 \end{array}}$

• Explain the FOIL method to Jack.

  Hints

  Here you see an example for $(2x+4)^2$ using the area model. This corresponds to the FOIL method.

  $(2x + 4)^2 = (2x)^2 + 2x(4) + 4(2x) + 4^2$

  Remember, you have to apply the exponent to every term in the parentheses.

  ${ \begin{array}{rcl} (3y)^2 &=&\\ &=& 3y(3y)\\ &=& 9y^2 \end{array}}$

  Look at this example for the FOIL method:

  ${ \begin{array}{rcl} (2a - b) &=&\\ &=& 2a(2a) + 2a(-b) + -b(2a) + -b(-b)\\ &=& (2a)^2 - 4ab + b^2 \end{array}}$

  Solution

  We can expand the given perfect square binomial, $(5x-2y)^2$, using the FOIL method. First, we transform the expression:

  ${ (5x-2y)^2=(5x-2y)(5x-2y)}$

  $~$

  Let's start.

  F multiply the first ${\large 5x(5x) = 25x^2}$

  O multiply the outer ${\large 5x(-2y) = -10xy}$

  I multiply the inner ${\large -2y(5x) = -10xy}$

  L multiply the last ${\large -2y(-2y) = 4y^2}$

  $~$

  We still have to combine like terms.

  ${(5x - 2y)^2 = 25x^2 - 20xy + 4y^2}$

  Party on, Jack!

• Explain the FOIL method.

  Hints

  You can use the distributive property for ${ a(a-b)=a^2-ab}$ and for ${ -b(a-b)=-ab+b^2}$.

  Here is another example.

  Solution

  We can simplify ${ (a - b)^2 = (a - b)(a - b)}$ using FOIL multiplication.

  For ${ (a + b)(a + b)}$, use the FOIL method:

  F multiply the first ${ a \times a}$

  O multiply the outer ${ a \times -b}$

  I multiply the inner ${ -b \times a}$

  L multiply the last ${ -b \times b}$

  For the final step, we add all the resulting terms:

  ${ a^2 - ab - ab + b^2 = a^2 - 2ab + b^2}$

• Simplify the following expressions.

  Hints

  Use the formulas above and think about what to assign to $a$ as well as to $b$.

  For example: $(2x-4)$.

  You use the second formula with $a = 2x$ and b = 4$.

  You can use those formulas also for calculating products of numbers

  ${ 22 \times 18 = (20 + 2)(20 - 2)}$.

  Now we can plug in ${ a = 20}$ and ${ b = 2}$ in our third formula.

  ${ \begin{array}{rcl} 22 \times 18 &=& (a + b)(a - b)\\ &=& a^2 - b^2\\ &=& 20^2 - 2^2\\ &=& 400 - 4\\ &=& 396 \end{array}}$

  Solution

  These formulas show a pattern to simplify the multiplication of special binomials.

  Now, we'd like to practice using these formulas by substituting ${a}$ and ${ b}$ in the expressions:

  First expression: $(12 - 3z)^2$
  When ${ a = 12}$ and ${ b = 3z}$, we can use the second equation, ${(a + b)^2}$.

  ${ \begin{array}{rcl} (12 - 3z)^2 &=&\\ &=& 12^2 - 2(12)(3z) + (3z)^2\\ &=& 144 - 72z + 9z^2 \end{array}}$

  $~$

  Second expression: $96 \times 104$
  We should recognize that we can use the third formula:

  ${ \begin{array}{rcl} 96 \times 104 &=& (a + b)(a - b) \end{array}}$

  Substituting $a = 100$ and $b = 4$, we get:

  ${ \begin{array}{rcl} &=& (100 - 4)(100 + 4)\\ &=& (100 + 4)(100 - 4)\\ &=& 100^2 - 4^2\\ &=& 10000 - 16\\ &=& 9984 \end{array}}$

  $~$

  Third expression: $(1 + 9z)^2$
  Using the second formula with $a = 1$ and $b = 9z$, we get:

  ${ \begin{array}{rcl} (1 + 9z)^2 &=&\\ &=& 1^2 + 2(1)(9z) + (9z)^2\\ &=& 1 + 18z + 81z^2 \end{array}}$

  $~$

  Fourth expression: $(-3z - 4)^2$
  Given the expression $(-3y - 4)^2$, we can use the first formula, but we have to remember to pay attention to the signs.

  Using $a = -3y$ and $b = -4$, we get:

  ${ \begin{array}{rcl} (-3z - 4)^2 &=&\\ &=& (-3z)^2 + 2(-3z)(-4) + (-4)^2\\ &=& 9z^2 - 24z + 16 \end{array}}$