A polynomial subtraction is an operation between algebraic expressions. To subtract polynomials we must subtract the coefficients of like terms, grouping the terms with the same literal part. Note that the logic that accompanies this process is also used in adding polynomials.
For example, the difference between the polynomial \(2x^2-6x\) and the polynomial \(x^2+3\) It is \((2-1) x^2+(-6-3)x=x^2-9x\).
Read too: How to factor polynomials?
Topics of this article
Summary on subtracting polynomials
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Polynomial subtraction is an operation that groups terms with the same literal part.
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It is essential to know and apply the rule of signs in the subtraction of polynomials.
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If the polynomials have different degrees, the “missing” terms can be expressed by a null coefficient.
How to subtract polynomials?
To subtract polynomials we need to subtract the coefficients of the like terms, that is, of the terms with the same literal part. In other words, if p and q are polynomials and we seek \(because\), we must subtract the p-independent term from the q-independent term, then subtract the coefficient of the x-term of p from the coefficient of the x-term of q, and so on for all p and q terms. It is important to note that the operation of subtracting polynomials follows the same idea as adding polynomials.
Observation: In this text we will use powers of x to indicate the literal part of polynomials, but a polynomial can have other letters in the literal part.
Let’s look at the formal representation of the subtraction of polynomials before checking some examples.
Consider two polynomials of degree n, represented by
\(a_0+a_1 x+a_2 x^2+a_3 x^3+⋯+a_n x^n\)
\(b_0+b_1 x+b_2 x^2+b_3 x^3+⋯+b_n x^n\)
A subtraction between the two polynomials is given by:
\((a_0-b_0 )+(a_1-b_1 )x+(a_2-b_2 ) x^2+(a_3-b_3 ) x^3+⋯+(a_n-b_n ) x^n\)
Note that this expression is obtained using the sign rule:
\((a_0+a_1 x+a_2 x^2+a_3 x^3+⋯+a_n x^n )-(b_0+b_1 x+b_2 x^2+b_3 x^3+⋯+b_n x^n )\ )
\(=a_0+a_1 x+a_2 x^2+a_3 x^3+⋯+a_n x^n-b_0-b_1 x-b_2 x^2-b_3 x^3-…-b_n x^n\)
\(=\color{red}{a_0-b_0}+\color{blue}{a_1 x-b_1 x}+\color{green}{a_2 x^2-b_2 x^2}+\color{purple}{ a_3 x^3-b_3 x^3}+⋯+\color{orange}{a_n x^n-b_n x^n}\)
\(=\color{red}{(a_0-b_0 )}+\color{blue}{(a_1-b_1 )x}+\color{green}{(a_2-b_2 ) x^2}+\color{purple }{(a_3-b_3 ) x^3}+⋯+\color{orange}{(a_n-b_n ) x^n}\)
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Observation 1: The coefficients are real numbers and, therefore, can assume positive or negative values. Consequently, it is necessary to use the sign rule when subtracting polynomials.
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Observation two: If one or more terms are expressed in only one of the polynomials, we can represent them in the other polynomial with a null coefficient. For example, the polynomial \(5x^3 \) can be written as \(5x^3+0x^2+0x+0\). This type of representation is particularly important in the construction of a subtraction of polynomials, as we will see below.
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Examples of subtracting polynomials
Note that this is equivalent to “distributing” the subtraction symbol, i.e. applying the sign rule:
\((9x+1)\color{red}{-(7x-2)}= 9x + 1 \color{red}{-7x +2} = 9x-7x +1 +2 = 2x+3\)
Important: Notice that, due to the sign rule, the sign of each term of the second polynomial changes.
\(=\color{blue}3x^2\color{green}{-6}x\color{orange}{+9}\)
This is equivalent to
\((4x^2–6x+8)\color{red}{-(x^2-1)}=4x^2-6x+8\color{red}{-x^2+ 1}\)
\(=4x^2-x^2-6x+8+1=3x^2-6x+9\)
Read too: How to do polynomial division?
Solved exercises on subtraction of polynomials
question 1
Please rate each statement below as T (True) or F (False).
I. In the subtraction of polynomials it is not necessary to use the rule of signs.
II. If p is a polynomial of degree n and q is a polynomial of degree m, with n > m, then the polynomial \(because\) it is grade n.
III. \( (x^3+2x)-(x^2+3)=x-1\)
The correct order, from top to bottom, is
a) VVV
b) FVF
c) ELV
d) VFF
e) FFF
Resolution
I. False. The rule of signs is fundamental in subtracting polynomials.
II. True.
III. Fake. \( (x^3+2x)-(x^2+3)=x^3-x^2+2x-3\)
Alternative B.
question 2
if \((rx^5-3x^2+x)-(x^5+sx^2-4x)=7x^5-12x^2+5x\)then the values of res are, respectively, equal to
a) -8 and 5
b) 9 and 7
c) -9 and 5
d) -8 and -9
e) 8 and 9
Resolution
note that
\((rx^5-3x^2+x)\color{red}{-(x^5+sx^2-4x)}=rx^5-3x^2+x\color{red}{-x ^5-sx^2+4x}\)
Note that, from the difference between the polynomials, we can compare the coefficients:
\(rx^5-x^5=7x^5\)
\((r-1) x^5=7x^5\)
\(r – 1 = 7\)
\(r = 8\)
\(-3x^2-sx^2=-12x^2\)
\((-3-s) x^2=-12x^2\)
\(-3-s = -12\)
\(s = 9\)
By Maria Luiza Alves Rizzo
Math teacher