Subtracting Polynomials

Subtracting Polynomials exercise

• Explain how to calculate the amount of material needed for the frame.

  Hints

  Instead of subtracting a term, you can also add the negative of a term.

  If you have a parentheses with a minus in front, you should multiply each term inside the parentheses by $-1$.

  Remember, you can only combine like terms.

  Solution

  To find out how much material Ross needs for the frame, he has to subtract $x^2-2x-8$ from $2x^2-8x$.

  To do this, we can use the vertical function addition method:

  First, we write both equations, one below the other:

  $\begin{array}{rr} y_1=&2x^2-8x~~~~~\,~~~\\ -y_2=&-(~~x^2-2x-8) \end{array}$

  Instead of subtracting the term inside the parentheses, we can also add the negative of this term:

  $\begin{array}{rr} y_1=&2x^2-8x~~~~~~~\\ -y_2=&-x^2+2x+8 \end{array}$

  Now, we combine the like terms to get $y_1-y_2=x^2-6x+8$.

• Calculate the difference with the vertical function addition method.

  Hints

  Keep the distributive property in mind:

  $-(a+b-c)=-a-b+c$

  Multiplying an entire expression by $-1$ is the same as changing the sign of each term inside the parentheses.

  Here are some examples for combining like terms:

  • $2x+4x=6x$
  • $3-x^2+x^2=3$
  • $2x-4+4x+3=6x-1$

  You can add or subtract like terms by adding or subtracting the coefficients of those terms.

  Solution

  To subtract polynomials we can use the vertical function addition method. This means that we write the polynomial one below the other.

  $\begin{array}{l} ~~~~\,3x^2+2\\ -(~~x^2-2x+4) \end{array}$

  Next, we change the sign of the polynomial we want to subtract.

  $\begin{array}{l} ~3x^2+2\\ -x^2+2x-4 \end{array}$

  Now, we rearrange the terms in such a way that the like terms are lined up vertically.

  $\begin{array}{r} 3x^2~~~~~~~~~+2\\ -x^2+2x-4 \end{array}$

  Finally, we can combine the like terms and get the result $2x^2+2x-2$.

• Examine the variables in the expressions and determine whether or not you can subtract them from the given expressions.

  Hints

  You can only combine like-terms. These are terms for which the variable combination, as well as the degree of each variable are identical.

  Keep the following in mind:

  When subtracting composed terms such as $2x^3y^3$, the variables should be identical.

  For example, the variable pair for $2x^3y^3$ is $x^3y^3$.

  We cannot subtract a portion of the variable pair.

  So $x^3y^3 - x \ne x^2y^3$.

  Solution

  It's very important to keep in mind that you can only combine like-terms.

  As a special example, we have the combination of variables $y^2x$. It's absolutely impossible to add or subtract a portion of the variables, such as $y^2$, $y$ or $x$.

  $\mathbf{y^2x+3x}$

  • $5x$
  • $x-2y^2x$
  • $4y^2x$

  $\mathbf{3y^2-4xy+y}$

  • $7y^2$
  • $y-xy$
  • $-y^2+2y$

  All the other terms may not be subtracted from either polynomial.

• Determine the equation for the border of the flower patch.

  Hints

  To write an expression in standard form, arrange the terms according to degree, decreasing from left to right.

  You can add or subtract like-terms by adding or subtracting their coefficients.

  You can use the horizontal or the vertical method of combining polynomials. It doesn't matter which method you use, the result is always the same.

  Solution

  Here you can see how to subtract polynomials using the horizontal method. You can use the vertical method as well.

  First, we write each equation in standard form.

  $5x+3x^2=3x^2+5x$

  $2x^2-3+2x=2x^2+2x-3$

  Now we have

  $(3x^2+5x)-(2x^2+2x-3) = 3x^2+5x-2x^2-2x+3$

  Next we combine the like-terms.

  $3x^2+5x-2x^2-2x+3 = (3x^2-2x^2)+(5x-2x)+3$

  Finish it off with PEMDAS.

  $(3x^2-2x^2)+(5x-2x)+3 = x^2+3x+3$

• Analyze each statement about subtracting polynomials.

  Hints

  Here, you can see an example of how to write a polynomial in standard form.

  The different terms are arranged in decreasing degree, from left to right.

  You can imagine the combination of terms as the following:

  • Two apples plus three apples are $2+3=5$ apples.
  • You can't add two apples to three pears.

  Above is an example of the horizontal method for subtracting two polynomials.

  Solution

  What do you have to keep in mind when subtracting polynomials?

  • Write each polynomial in standard form: Arrange the terms according to degree, decreasing from left to right.

  • You can only combine like-terms.: There is no way to combine the terms $2x$ and $-5y^2$, for instance.

  • Composed terms: When subtracting composed terms, such as $3xy^2$, the variables should be identical. For example, the variable pair for $3xy^2$ is $xy^2$. We cannot subtract a portion of the variable pair. So $xy^2 - x \ne y^2$.

• Solve the following subtraction problems.

  Hints

  You can use the vertical or horizontal method for combining polynomial expressions.

  Remember to only combine like terms.

  Like terms are terms in which the variable combination as well as the degree of the exponents are identical.

  Here you can see an example of how to combine like terms.

  $\begin{array}{rcl} 7xy-3x+5x-2xy&=&(7xy-2xy)+(-3x+5x)\\ &=&5xy+2x \end{array}$

  Solution

  The vertical method of combining polynomials will be used for the first two examples and the horizontal method will be used for the remaining two.

  ${(2xy+3x-4)-(3x^2+2xy-6)}$

  $\begin{array}{rcccccccc} &&&2xy&+&3x&-&4\\ &-&(&3x^2&+&2xy&-&6&) \end{array}$

  First, we change the sign of the inferior polynomial and rearrange the monomials.

  $\begin{array}{rcccccccc} &&&2xy&&+&3x&-&4\\ -&3x^2&-&2xy&&&&-&6& \end{array}$

  Finally, we combine the like terms to $-3x^2+3x-10$.

  $~$

  ${(2x^2+3xy^2-4x)-(3x^2y+4x^2-6x)}$

  $\begin{array}{rccccccccccc} &&2x^2&+&3xy^2&-&4x&\\ &-&3x^2y&-&4x^2&+&6x& \end{array}$

  Again, we change the sign of the inferior polynomial and rearrange the monomials.

  $\begin{array}{rccccccccccc} &&&&3xy^2&+&2x^2&-&4x&\\ &-&3x^2y&&&-&4x^2&+&6x& \end{array}$

  Combining like terms, we get $-3x^2y+3xy^2-2x^2+2x$.

  $~$

  ${(3x^2y^2+3x^2y-4xy^2+4x)-(6xy^2-4x^2y+2x^2)}$

  First, we arrange the second term in standard form.

  $6xy^2-4x^2y+2x^2=-4x^2y+6xy^2+2x^2$

  $(3x^2y^2+3x^2y-4xy^2+4x)-(6xy^2-4x^2y+2x^2)$ $=3x^2y^2+3x^2-4y^2+4x+4x^2y-6xy^2-2x^2$

  Now we combine the like terms.

  $3x^2y^2+3x^2y-4xy^2+4x+4x^2y-6xy^2-2x^2$ $=3x^2y^2+(3x^2y+4x^2y)+(-4xy^2-6xy^2)-2x^2+4x$.

  Almost done, just one more step.

  $3x^2y^2+(3x^2y+4x^2y)+(-4xy^2-6xy^2)-2x^2+4x$ $=3x^2y^2+7x^2y-10xy^2-2x^2+4x$

  $~$

  ${(12x+4xy^2+3x^2+7)-(3x^2+7xy-5x+6)}$

  First, arrange the first expression in standard form.

  $12x+4xy^2+3x^2+7=4xy^2+3x^2+12x+7$

  $(4xy^2+3x^2+12x+7)-(3x^2+7xy-5x+6)=$

  Now, combine the like terms.

  $(4xy^2+3x^2+12x+7)-(3x^2+7xy-5x+6)$ $=4xy^2+(3x^2-3x^2)-7xy+(12x+5x)+(7-6)$.

  The last step!

  $4xy^2+(3x^2-3x^2)-7xy+(12x+5x)+(7-6)=4xy^2-7xy+17x+1$