Adding Polynomials

Adding Polynomials exercise

• Find out how much space the tents and the stage take up by adding the polynomials.

  Hints

  To get the standard form of a polynomial arrange the terms depending on their degree.

  Start with the highest degree.

  Here you have an example.

  You can only combine like terms.

  When adding or subtracting like terms, you have to add or subtract the coefficients.

  $\begin{array}{rcr} 2x^2+5x^2-3x^2 &=& \\ (2+5-3)x^2 &=& \\ &=& 4x^2 \end{array}$

  Solution

  $5+3x+\frac12x^2$ as well as $3x^2+4-3x$ are polynomials.

  In this exercise, you were asked to add the polynomials horizontally. To do so, the first step is always rearranging the monomials in decreasing order of degree from left to right:

  • $5+3x+\frac12x^2=\frac12x^2+3x+5$
  • $3x^2+4-3x=3x^2-3x+4$

  Now, we regroup the polynomials by writing all like terms in the same sets of parentheses:

  $\left(\frac12x^2+3x^2\right)+(3x-3x)+(5+4)$

  Now we add or subtract the like terms by adding or subtracting the coefficients. We get:

  $3\frac12x^2+9$

• Add the polynomials vertically to find the combined space needed for tents and for the stage.

  Hints

  When adding or subtracting like terms, you add or subtract the coefficients:

  $7x+3x=(7+3)x=10x$

  Like terms are like fruits: If you add $3$ apples and $2$ bananas, you will end up with $3$ apples and $2$ bananas, not $5$ app-nanas.

  Solution

  When adding polynomials, you first have to arrange the monomials in standard form in decreasing order of degree from left to right.

  Next, write each monomial with like terms below each other. If you have terms in one polynomial where there is no matching term in the second polynomial, leave some space in the line of the second polynomial, and vice versa.

  So when adding $\frac12 x^2+3x+5$ and $3x^2-3x+4$ vertically, you can write it like this:

  $~~~~~\frac12 x^2+3x+5$

  $~+3x^2-3x+4$

  Next, you combine all like terms:

  • $\frac12x^2+3x^2=\left(\frac12+3\right)x^2=3\frac12x^2$
  • $3x-3x=(3-3)x=0x=0$
  • $4+5=9$

  So we get:

  $y_1+y_2=3\frac12x^2+0x+9=3\frac12x^2+9$

• Identify the degrees of the given monomials.

  Hints

  The degree of a monomial doesn't depend on the coefficient.

  The degree of $x^5$ is its exponent, $5$.

  The degree of $7x^2y^4$ is $2+4=6$.

  The degree of any constant is $0$.

  The degree of $x$ is $1$.

  Solution

  Each monomial is assigned a degree depending on the exponent of its variable. If the monomial contains more than one variable, the exponents are added. The degree of any constant is $0$, whereas the degree of $x$ is $1$.

  Here, you see the correct order for the given monomials.

  1. $x^3$ (degree: $3$), $4xy$ (degree: $1+1=2$), $5y$ (degree: $1$), $7$ (degree: $0$)
  2. $3x^3y^2$ (degree: $3+2=5$), $x^4$ (degree: $4$), $2xy^2$ (degree: $1+2=3$), $7x$ (degree: $1$)
  3. $2x^3$ (degree: $3$), $3x^2$ (degree: $2$), $x$ (degree: $1$), $2$ (degree: $0$)

• Calculate the total space needed for the birthday party by adding the two polynomials.

  Hints

  Fill in the coefficients as well as the operation signs, $+$ or $-$.

  First, write the polynomials in standard form.

  $-3+5x^2+x \rightarrow 5x^2+x-3$

  Remember to only combine like terms.

  You can add like terms by adding the coefficients.

  $2x^3-4x^3=(2-4)x^3=-2x^3$

  Solution

  We have to add $y_1=2x-2+2x^2$ and $y_2=1+2x^2-4x$.

  First, we arrange both polynomials in standard form according to the degrees of the monomials:

  $\begin{array}{lccccccc} y_1=&2&x^2&+&2&x&-&2\\ y_2=&2&x^2&-&4&x&+&1\\ \hline &4&x^2&-&2&x&-&1 \end{array}$

  In the last row, we add all like terms together by adding the coefficients.

• Determine the degrees of the monomials.

  Hints

  Remember that sometimes terms have hidden variables and/or exponents, like $x^0=1$ and $x^1=x$.

  For each monomial with more than one variable, the exponents are added:

  The degree of $12xy^3$ can be found by adding the exponents of $x$ and $y$: $1+3=4$.

  Solution

  The degree of any polynomial is the highest degree monomial in the polynomial.

  If we have a monomial with more than one variable, we add the exponents to determine its degree.

  Remember, $x=x^1$ and $x^0=1$, so for example, $3x=3x^1$; therefore, its degree is $1$. Additionally, in $4=4x^0$, the degree is $0$.

  So we have the following degrees:

  1. The monomial $2x^2$ has the degree $2$.
  2. The monomial $3x^4y^2$ has the degree $4+2=6$.
  3. The monomial $5x=5x^1$ has the degree $1$.
  4. The monomial $3=3x^0$ has the degree $0$.

• Simplify the given polynomials by combining all like monomials.

  Hints

  You can only combine like terms.

  For like terms, the variables and the exponents match.

  Group like terms in sets of parentheses when adding horizontally or below each other when adding vertically.

  Solution

  In math, you often have to simplify. This means you have to combine like terms.

  Like terms contain the same variable as well as the same exponent.

  For example, $x^3$ an $5x^3$ are like terms. You can add them together and get $6x^3$.

  On the other hand, $3x^2$ and $2x^3$ aren't like terms, their variables match, but the exponents don't. $3x^2$ and $3y^2$ aren't like term either, their variables don't match.

  $\begin{array}{rcl} ~\\ a^2x^2-2ax^2+3a^2x-5ax^2+2a^2x^2 &=&(a^2x^2+2a^2x^2)+3a^2x+(-2ax^2-5ax^2)\\ &=& 3a^2x^2+3a^2x-7ax^2\\ ~\\ \end{array}$

  $\begin{array}{rcl} ~\\ ax^2-2ax+3a^2x^2-5a^2x+2ax^2 &= & 3a^2x^2-5a^2x+(ax^2+2ax^2)-2ax\\ &=&3a^2x^2-5a^2x+3ax^2-2ax\\ ~\\ \end{array}$

  $\begin{array}{rcl} ~\\ ax-2a^2x+3ax^2-5a^2x^2+2a^2x &= &-5a^2x^2+(-2a^2x^2+2a^2x)+3ax^2+ax\\ &=&-5a^2x^2+3ax^2+ax\\ \end{array}$