6.1 Add and subtract polynomials (Page 2/12)

Page 2 / 12

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

This table has 11 rows and 5 columns. The first column is a header column, and it names each row. The first row is named “Monomial,” and each cell in this row contains a different monomial. The second row is named “Degree,” and each cell in this row contains the degree of the monomial above it. The degree of 14 is 0, the degree of 8y squared is 2, the degree of negative 9x cubed y to the fifth power is 8, and the degree of negative 13a is 1. The third row is named “Binomial,” and each cell in this row contains a different binomial. The fourth row is named “Degree of each term,” and each cell contains the degrees of the two terms in the binomial above it. The fifth row is named “Degree of polynomial,” and each cell contains the degree of the binomial as a whole.” The degrees of the terms in a plus 7 are 0 and 1, and the degree of the whole binomial is 1. The degrees of the terms in 4b squared minus 5b are 2 and 1, and the degree of the whole binomial is 2. The degrees of the terms in x squared y squared minus 16 are 4 and 0, and the degree of the whole binomial is 4. The degrees of the terms in 3n cubed minus 9n squared are 3 and 2, and the degree of the whole binomial is 3. The sixth row is named “Trinomial,” and each cell in this row contains a different trinomial. The seventh row is named “Degree of each term,” and each cell contains the degrees of the three terms in the trinomial above it. The eighth row is named “Degree of polynomial,” and each cell contains the degree of the trinomial as a whole. The degrees of the terms in x squared minus 7x plus 12 are 2, 1, and 0, and the degree of the whole trinomial is 2. The degrees of the terms in 9a squared plus 6ab plus b squared are 2, 2, and 2, and the degree of the trinomial as a whole is 2. The degrees of the terms in 6m to the fourth power minus m cubed n squared plus 8mn to the fifth power are 4, 5, and 6, and the degree of the whole trinomial is 6. The degrees of the terms in z to the fourth power plus 3z squared minus 1 are 4, 2, and 0, and the degree of the whole trinomial is 4. The ninth row is named “Polynomial,” and each cell contains a different polynomial. The tenth row is named “Degree of each term,” and the eleventh row is named “Degree of polynomial.” The degrees of the terms in b plus 1 are 1 and 0, and the degree of the whole polynomial is 1. The degrees of the terms in 4y squared minus 7y plus 2 are 2, 1, and 0, and the degree of the whole polynomial is 2. The degrees of the terms in 4x to the fourth power plus x cubed plus 8x squared minus 9x plus 1 are 4, 3, 2, 1, and 0, and the degree of the whole polynomial is 4.

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

Find the degree of the following polynomials.

ⓐ $10 y$
ⓑ $4 x^{3} - 7 x + 5$
ⓒ $−15$
ⓓ $−8 b^{2} + 9 b - 2$
ⓔ $8 x y^{2} + 2 y$

Solution

ⓐ
$\begin{matrix} 10 y \\ The exponent of y is one. y = y^{1} & The degree is 1. \end{matrix}$
ⓑ
$\begin{matrix} 4 x^{3} - 7 x + 5 \\ The highest degree of all the terms is 3. & The degree is 3. \end{matrix}$
ⓒ
$\begin{matrix} −15 \\ The degree of a constant is 0. & The degree is 0. \end{matrix}$
ⓓ
$\begin{matrix} −8 b^{2} + 9 b - 2 \\ The highest degree of all the terms is 2. & The degree is 2. \end{matrix}$
ⓔ
$\begin{matrix} 8 x y^{2} + 2 y \\ The highest degree of all the terms is 3. & The degree is 3. \end{matrix}$

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Find the degree of the following polynomials:

ⓐ $−15 b$ ⓑ $10 z^{4} + 4 z^{2} - 5$ ⓒ $12 c^{5} d^{4} + 9 c^{3} d^{9} - 7$ ⓓ $3 x^{2} y - 4 x$ ⓔ $−9$

ⓐ $1$ ⓑ $4$ ⓒ $12$ ⓓ 3 ⓔ 0

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Find the degree of the following polynomials:

ⓐ $52$ ⓑ $a^{4} b - 17 a^{4}$ ⓒ $5 x + 6 y + 2 z$ ⓓ $3 x^{2} - 5 x + 7$ ⓔ $− a^{3}$

ⓐ $0$ ⓑ $5$ ⓒ $1$ ⓓ 2 ⓔ 3

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Add and subtract monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

Add: $25 y^{2} + 15 y^{2}$ .

Solution

$\begin{matrix} 25 y^{2} + 15 y^{2} \\ Combine like terms. & 40 y^{2} \end{matrix}$

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Add: $12 q^{2} + 9 q^{2} .$

$21 q^{2}$

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Add: $−15 c^{2} + 8 c^{2} .$

$−7 c^{2}$

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Subtract: $16 p - (−7 p)$ .

Solution

$\begin{matrix} 16 p - (−7 p) \\ Combine like terms. & 23 p \end{matrix}$

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Subtract: $8 m - (−5 m) .$

$13 m$

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Subtract: $−15 z^{3} - (−5 z^{3}) .$

$−10 z^{3}$

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Remember that like terms must have the same variables with the same exponents.

Simplify: $c^{2} + 7 d^{2} - 6 c^{2}$ .

Solution

$\begin{matrix} c^{2} + 7 d^{2} - 6 c^{2} \\ Combine like terms. & −5 c^{2} + 7 d^{2} \end{matrix}$

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Add: $8 y^{2} + 3 z^{2} - 3 y^{2} .$

$5 y^{2} + 3 z^{2}$

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Add: $3 m^{2} + n^{2} - 7 m^{2} .$

$−4 m^{2} + n^{2}$

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Simplify: $u^{2} v + 5 u^{2} - 3 v^{2}$ .

Solution

$\begin{matrix} u^{2} v + 5 u^{2} - 3 v^{2} \\ There are no like terms to combine. & u^{2} v + 5 u^{2} - 3 v^{2} \end{matrix}$

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Simplify: $m^{2} n^{2} - 8 m^{2} + 4 n^{2} .$

There are no like terms to combine.

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Simplify: $p q^{2} - 6 p - 5 q^{2} .$

There are no like terms to combine.

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Add and subtract polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Find the sum: $(5 y^{2} - 3 y + 15) + (3 y^{2} - 4 y - 11) .$

Solution

Identify like terms.
Rearrange to get the like terms together.
Combine like terms.

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Find the sum: $(7 x^{2} - 4 x + 5) + (x^{2} - 7 x + 3) .$

$8 x^{2} - 11 x + 1$

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Find the sum: $(14 y^{2} + 6 y - 4) + (3 y^{2} + 8 y + 5) .$

$17 y^{2} + 14 y + 1$

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Find the difference: $(9 w^{2} - 7 w + 5) - (2 w^{2} - 4) .$

Solution


Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

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Find the difference: $(8 x^{2} + 3 x - 19) - (7 x^{2} - 14) .$

$15 x^{2} + 3 x - 5$

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Find the difference: $(9 b^{2} - 5 b - 4) - (3 b^{2} - 5 b - 7) .$

$6 b^{2} + 3$

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Subtract: $(c^{2} - 4 c + 7)$ from $(7 c^{2} - 5 c + 3)$ .

Solution



Distribute and identify like terms.
Rearrange the terms.
Combine like terms.

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Subtract: $(5 z^{2} - 6 z - 2)$ from $(7 z^{2} + 6 z - 4)$ .

$2 z^{2} + 12 z - 2$

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Subtract: $(x^{2} - 5 x - 8)$ from $(6 x^{2} + 9 x - 1)$ .

$5 x^{2} + 14 x + 7$

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Find the sum: $(u^{2} - 6 u v + 5 v^{2}) + (3 u^{2} + 2 u v)$ .

Solution

$\begin{matrix} (u^{2} - 6 u v + 5 v^{2}) + (3 u^{2} + 2 u v) \\ Distribute. & u^{2} - 6 u v + 5 v^{2} + 3 u^{2} + 2 u v \\ Rearrange the terms, to put like terms together. & u^{2} + 3 u^{2} - 6 u v + 2 u v + 5 v^{2} \\ Combine like terms. & 4 u^{2} - 4 u v + 5 v^{2} \end{matrix}$

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Find the sum: $(3 x^{2} - 4 x y + 5 y^{2}) + (2 x^{2} - x y)$ .

$5 x^{2} - 5 x y + 5 y^{2}$

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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