4.5: Combining Polynomials Using Addition and Subtraction
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Like Terms
Like Terms
Terms whose variable parts, including the exponents, are identical are called like terms. Like terms is an appropriate name since terms with identical variable parts and different numerical coefficients represent different amounts of the same quantity. As long as we are dealing with quantities of the same type we can combine them using addition and subtraction.
An algebraic expression can be simplified by combining like terms.
Sample Set A
Combine the like terms.
6 houses + 4 houses = 10 houses. 6 and 4 are the same type give 10 of that type.
6 houses + 4 houses+ 2 motels = 10 houses + 2 motels . 6 and 4 of the same type give 10 of that type. Thus, we have 10 of one type and 2 of another type.
Suppose we let the letter \(x\) represent "house." Then, \(6x+4x=10x\). 6 and 4 of the same type give 10 of that type.
Suppose we let \(x\) represent "house" and \(y\) represent "motel."
\(6x + 4x + 2y = 10x + 2y\)
Practice Set A
Like terms with the same numerical coefficient represent equal amounts of the same quantity.
Like terms with different numerical coefficients represent
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different amounts of the same quantity
Combining Like Terms
Since like terms represent amounts of the same quantity, they may be combined, that is, like terms may be added together.
Simplify each of the following polynomials by combining like terms.
\(2x + 5x + 3x\).
There are \(2x\)'s, then 5 more, then 3 more. This makes a total iof \(10x\)'s.
\(7x + 8y - 3x\).
From \(7x\)'s, we lose \(3x\)'s. This makes \(4x\)'s. The \(8y\)'s represent a quantity different from the \(x\)'s and therefore will not combine with them.
\(7x + 8y - 3x = 4x + 8y\).
\(4a^3 - 2a^2 + 8a^3 + a^2 - 2a^3\).
\(4a^3, 8a^3\), and \(-2a^3\) represent quantities of the same type.
\(4a^3 + 8a^3 - 2a^3 = 10a^3\).
\(-2a^2\) and \(a^2\) represent quantities of the same type.
\(-2a^2 + a^2 = -a^2\).
Thus,
\(4a^3 - 2a^2 + 8a^3 + a^2 - 2a^3 = 10a^3 - a^2\).
Practice Set B
Simplify each of the following expressions.
\(4y + 7y\)
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\(11y\)
\(3x + 6x + 11x\)
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\(20x\)
\(5a + 2b + 4a - b - 7b\)
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\(9a - 6b\)
\(10x^3 - 4x^3 + 3x^2 - 12x^3 + 5x^2 + 2x + x^3 + 8x\)
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\(-5x^3 + 8x^2 + 10x\)
\(2a^5 - a^5 + 1 - 4ab - 9 + 9ab - 2 - 3 - a^5\).
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\(5ab - 13\).
Simplifying Expressions Containing Parentheses
Simplifying Expressions Containing Parentheses
When parentheses occur in expressions, they must be removed before the expression can be simplified. Parentheses can be removed using the distributive property.
Distributive Property
Sample Set C
Simplify each of the following expressions by using the distributive property and combining like terms.
\(4x + 9(x^2 - 6x - 2) + 5\) Remove parentheses.
\(4x + 9x^2 - 54x - 18 + 5\) Combine like terms.
\(-50x + 9x^2 - 13\)
By convention, the terms in an expression are placed in descending order with the highest degree term appearing first. Numerical terms are placed at the right end of the expression. The commutative property of addition allows us to change the order of the terms.
\(9x^2 - 50x - 13\)
\(2+2[5+4(1+a)]\)
Eliminate the innermost set of parentheses first
\(2+2[5+4+4a]\)
By the order of operations, simplify inside the parentheses before multiplying (by the 2)
\(2 + 2[9 + 4a]\) Remove this set of parentheses.
\(2 + 18 + 8a\) Combine like terms.
\(20 + 8a\) Write in descending order.
\(8a + 20\)
\(x(x−3)+6x(2x+3)\)
Use the rule for multiplying powers with the same base.
\(x^2 - 3x + 12x^2 + 18x\) Combine like terms.
\(13x^2 + 15x\)
Practice Set C
Simplify each of the following expressions by using the distributive property and combining like terms.
\(4(x+6)+3(2+x+3x^2)-2x^2\)
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\(7x^2+7x+30\)
\(7(x+x^3)-4x^3-x+1+4(x^2-2x^3+7)\)
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\(-5x^3+4x^2+6x+29\)
\(5(a+2)+6a-7+(8+4)(a+3a+2)\)
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\(59a+27\)
\(x(x+3)+4x^2+2x\)
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\(5x^2+5x\)
\(a^3(a^2+a+5)+a(a^4+3a^2+4)+1\)
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\(2a^5+a^4+8a^3+4a+1\)
\(2[8-3(x-3)]\)
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\(-6x+34\)
\(x^2+3x+7[x+4x^2+3(x+x^2)]\)
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\(50x^2+31x\)
Exercises
For the following problems, simplify each of the algebraic expressions.
\(x+3x\)
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\(4x\)
\(4x + 7x\)
\(9a + 12a\)
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\(21a\)
\(5m - 3m\)
\(10x - 7x\)
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\(3x\)
\(7y - 9y\)
\(6k - 11k\)
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\(-5k\)
\(3a+5a+2a\)
\(9y+10y+2y\)
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\(21y\)
\(5m−7m−2m\)
\(h−3h−5h\)
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\(-7h\)
\(a+8a+3a\)
\(7ab+4ab\)
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\(11ab\)
\(8ax+2ax+6ax\)
\(3a^2 + 6a^2 + 2a^2\)
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\(11a^2\)
\(14a^2b + 4a^2b + 19a^2b\)
\(10y - 15y\)
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\(-5y\)
\(7ab−9ab+4ab\)
\(210ab^4 + 412ab^4 + 100a^4b\) (Look closely at the exponents.)
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\(622ab^4 + 100a^4b\)
\(5x^2y^0 + 3x^2y + 2x^2y + 1, y \not = 0\) (Look closely at the exponents.)
\(8w^2 - 12w^2 - 3w^2\)
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\(-7w^2\)
\(6xy−3xy+7xy−18xy\)
\(7x^3 - 2x^2 - 10x + 1 - 5x^2 - 3x^3 - 12 + x\)
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\(4x^3 - 7x^2 - 9x - 11\)
\(21y−15x+40xy−6−11y+7−12x−xy\)
\(1x+1y−1x−1y+x−y\)
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\(x-y\)
\(5x^2 - 3x - 7 + 2x^2 - x\)
\(-2z + 15z + 4z^3 + z^2 - 6z^2 + z\)
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\(2z^3 - 5z^2 + 16z\)
\(18x^2y - 14x^2y - 20x^2y\)
\(-9w^5 - 9w^4 - 9w^5 + 10w^4\)
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\(-18w^5 + w^4\)
\(2x^4 + 4x^3 - 8x^2 + 12x - 1 - 7x^3 - 1x^4 - 6x + 2\)
\(17d^3r + 3d^3r - 5d^3r + 6d^2r + d^3r - 30d^2r + 3 - 7 + 2\)
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\(16d^3r - 24d^2r - 2\)
\(a^0 + 2a^0 - 4a^0, a \not = 0\)
\(4x^0 + 3x^0 - 5x^0 + 7x^0 - x^0, x \not = 0\)
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\(8\)
\(2a^3b^2c + 3a^2b^2c^0 + 4a^2b^2 - a^3b^2c, c \not = 0\)
\(3z−6z+8z\)
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\(5z\)
\(3z^2 - z + 3z^3\)
\(6x^3 + 12x + 5\)
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\(6x^3 + 12x + 5\)
\(3(x+5)+2x\)
\(7(a+2)+4\)
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\(7a+18\)
\(y+5(y+6)\)
\(2b+6(3−5b)\)
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\(−28b+18\)
\(5a−7c+3(a−c)\)
\(8x−3x+4(2x+5)+3(6x−4)\)
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\(31x+8\)
\(2z+4ab+5z−ab+12(1−ab−z)\)
\((a+5)4+6a−20\)
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\(10a\)
\((4a+5b−2)3+3(4a+5b−2)\)
\((10x + 3y^2)4 + 4(10x + 3y^2)\)
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\(80x + 24y^2\)
\(2(x−6)+5\)
\(1(3x+15)+2x−12\)
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\(5x+3\)
\(1(2 + 9a + 4a^2) + a^2 - 11a\)
\(1(2x - 6b + 6a^2b + 8b^2) + 1(5x + 2b - 3a^2b\)
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\(3a^2b + 8b^2 - 4b + 7x\)
After observing the following problems, can you make a conjecture about \(1(a+b)\)?
\(1(a+b) =\)
Using the result of problem 52, is it correct to write
\((a+b)=a+b?\)
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yes
\(3(2a + 2a^2) + 8(3a + 3a^2)\)
\(x(x + 2) + 2(x^2 + 3x - 4\)
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\(3x^2 + 8x - 8\)
\(A(A+7) + 4(A^2 + 3a + 1)\)
\(b(2b^3 + 5b^2 + b + 6) - 6b^2 - 4b + 2\)
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\(2b^4 + 5b^3 - 5b^2 + 2b + 2\)
\(4a−a(a+5)\)
\(x - 3x(x^2 - 7x - 1)\)
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\(-3x^3 + 21x^2 + 4x\)
\(ab(a - 5) - 4a^2b + 2ab - 2\)
\(xy(3xy + 2x - 5y) - 2x^2y^2 - 5x^2y + 4xy^2\)
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\(x^2y^2 - 3x^2y - xy^2\)
\(3h[2h+5(h+2)]\)
\(2k[5k+3(1+7k)]\)
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\(52k^2 + 6k\)
\(8a[2a−4ab+9(a−5−ab)]\)
\(6{m + 5n[n + 3(n-1)] + 2n^2} - 4n^2 - 9m\)
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\(128n^2 - 90n - 3m\)
\(5[4(r−2s)−3r−5s]+12s\)
\(8{9[b - 2a + 6c(c + 4) - 4c^2] + 4a + b} - 3b\)
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\(144c^2 - 112a + 77b + 1728c\)
\(5[4(6x−3)+x]−2x−25x+4\)
\(3xy^2(4xy + 5y) + 2xy^3 + 6x^2y^3 + 4y^3 - 12xy^3\)
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\(18x^2y^3 + 5xy^3 + 4y^3\)
\(9a^3b^7(a^3b^5-2a^2b^2+6) - 2a(a^2b^7 - 5a^5b^{12} + 3a^4b^9) - a^3b^7\)
\(−8(3a+2)\)
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\(−24a−16\)
\(−4(2x−3y)\)
\(-4xy^2[7xy - 6(5-xy^2) + 3(-xy + 1) +1]\)
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\(-24x^2y^4 - 16x^2y^3 + 104xy^2\)
Exercises for Review
Simplify \((\dfrac{x^{10}y^8z^2}{x^2y^6})^3\)
Find the value of \(\dfrac{-3(4-9)-6(-3)-1}{2^3}\)
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\(4\)
Write the expression \(\dfrac{42x^2y^5z^3}{21x^4y^7}\) so that no denominator appears.
How many \((2a+5)\)'s are there in \(3x(2a+5)\)
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\(3x\)
Simplify \(3(5n + 6m^2) - 2(3n + 4m^2)\)