Skills Refresher:  Operations Using Polynomials (6.4 – 6.6) — Intermediate Algebra

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9 Skills Refresher: Operations Using Polynomials (6.4 – 6.6) — Intermediate Algebra

6.4 Basic Operations Using Polynomials

Many applications in mathematics have to do with what are called polynomials. Polynomials are made up of terms. Terms are a product of numbers and/or variables. For example, $5x$ , $2y^2$ , $-5$ , $ab^3c$ , and $x$ are all terms. Terms are connected to each other by addition or subtraction.

Expressions are often defined by the number of terms they have.

$\begin{array}{l} \text{A monomial has one term, such as } x, xy, 3x^2.\\ \text{A binomial has two terms connected by a + or -, such as } a^2-b^2, 3x-y, 4x+2xy^3 \\ \text{A trinomial has three terms connected by a + or -, such as } ax^2 + bx + c \end{array}$

The term polynomial is generic for many terms. Monomials, binomials, trinomials, and expressions with more terms all fall under the umbrella of “polynomials.”

Polynomials are classified by the sum of exponents of the term with the highest exponent sum, which is called the degree of the polynomial.

A first degree polynomial is a linear polynomial and would not have any terms with a sum of exponents greater than one. Examples include $3x + 2y + 4z$ , $42x - 56y$ , $22z$ .

A second degree polynomial is a quadratic polynomial and would not have any terms with a sum of exponents greater than two. Examples include $3x^2 + 2x + 4y^2$ , $42xy - 3z$ , $2zx$ .

A third degree polynomial is a cubic polynomial and would not have any terms with a sum of exponents greater than three. Examples include $5x^2 + 2xy^2 + 4y^2$ , $42xyz - 5z^2$ , $3zx^2$ .

A fourth degree polynomial is a quartic polynomial and would not have any terms with a sum of exponents greater than four. Examples include $5x^2 + 3xy^2z + 4y^2$ , $42xyz - 6z^4$ , $8z^3x$ .

A fifth degree polynomial is a quintic polynomial.

A sixth degree polynomial is a sextic polynomial.

A seventh degree polynomial is a septic polynomial.

A eighth degree polynomial is a octic polynomial.

A ninth degree polynomial is a nonic polynomial.

A tenth degree polynomial is a decic polynomial.

The degree of any term is the sum of its exponents:

$x^9, x^7y^2, x^3y^3z^3$ are all ninth degree terms (nonic polynomial)

$x^6, x^4y^2, x^3yz^2$ are all sixth degree terms (sextic polynomial)

$x^4, x^2y^2, xyz^2$ are all fourth degree terms (quartic polynomial)

Terms of a polynomial are named in the order of their appearance. For instance, the polynomial $2x^4y + x^3y^2 - x^2y^3 + 5xy^4$ has four terms, each one of the fifth degree. The terms are numbered in order for this polynomial, starting from the first term $(2x^4y)$ and continuing to the second ((x^3y^2)\), third $(-x^2y^3)$ , and fourth terms $(5xy^4)$ .

If it is known what the variable in a polynomial represents, it is possible to substitute in the value and evaluate the polynomial, as shown in the following example.

Example 6.4.1

Simplify the expression $2x^2 - 4x + 6$ using $x = -4$ .

$\begin{array}{rlllll} \text{When } x=-4, \text{ replace all }x\text{ with }-4:&2x^2&-&4x&+&6 \\ \text{Reduce the exponents:}&2(-4)^2&-&4(-4)&+&6 \\ \text{Multiply all coefficients:}&2(16)&-&4(-4)&+&6 \\ \text{Add:}&32&+&16&+&6 \\ \text{Solution:}&54&&&& \end{array}$

Remember the exponent only affects the number to which it is physically attached. This means −3² = −9 because the exponent is only attached to the 3, whereas (−3)² = 9 because the exponent is attached to the parentheses and everything inside.

Example 6.4.2

Simplify the expression $-x^2 + 2x + 6$ using $x = 3$ .

$\begin{array}{rlllll} \text{When }x=3,\text{ replace all }x \text{ with 3:}&-x^2&+&2x&+&6 \\ \text{Reduce the exponents:}&-(3)^2&+&2(3)&+&6 \\ \text{Multiply:}&-9&+&2(3)&+&6 \\ \text{Add:}&-9&+&6&+&6 \\ \text{Solution:}&3&&&& \end{array}$

Sometimes, when working with polynomials, the value of the variable is unknown and the polynomial will be simplified rather than ending with some value. The simplest operation with polynomials is addition. When adding polynomials, it is merely combining like terms. Consider the following example.

Example 6.4.3

Simplify the expression $(4x^3 - 2x + 8) + (3x^3 - 9x^2 - 11).$

The first thing one should do is place the equations one over top of the other, ordering each of the terms into columns so they can simply be added or subtracted.

$\begin{array}{rrrrrrrl} 4x^3&&&-&2x&+&8& \\ 3x^3&-&9x^2&&&-&11&\text{Add each of the columns} \\ \midrule 7x^3&-&9x^2&-&2x&-&3&\text{Solution} \\ \end{array}$

Subtracting polynomials is almost as fast as addition. Subtraction is generally shown by a minus sign in front of the parentheses. When there is a negative sign in front of parentheses, distribute it throughout the expression, changing the signs of everything inside.

Example 6.4.4

Simplify the expression $(5x^2 - 2x + 7) - (3x^2 + 6x - 4).$

$\begin{array}{rrrrrrl} &5x^2&-&2x&+&7& \\ -&(3x^2&+&6x&-&4)&\text{Subtract each bottom term from each top term} \\ \midrule &2x^2&-&8x&+&11&\text{Solution} \end{array}$

Example 6.4.5

Simplify the expression $(2x^2 - 4x + 3) + (5x^2 - 6x + 1) - (x^2 - 9x + 8).$

$\begin{array}{rrrrrrl} &2x^2&-&4x&+&3& \\ &5x^2&-&6x&+&1& \\ -&(x^2&-&9x&+&8)&\text{Add and subtract} \\ \midrule &6x^2&-&x&-&4&\text{Solution} \\ \end{array}$

6.5 Multiplication of Polynomials — Intermediate Algebra

Multiplying monomials is done by multiplying the numbers or coefficients and then adding the exponents on like factors. This is shown in the next example.

Example 6.5.1

Find the following product: $(4x^3y^4z)(2x^2y^6z^3).$

$\begin{array}{rrl} &4x^3y^4z& \\ \times & 2x^2y^6z^3&\text{Multiply coefficients and add exponents} \\ \midrule &8x^5y^{10}z^4&\text{Solution} \end{array}$

Some notes: $z$ has an exponent of 1 when no exponent is written. When adding or subtracting, the exponents will stay the same, but when multiplying (or dividing), the exponents will change.

Next, consider multiplying a monomial by a polynomial. We have seen this operation before with distributing throughout parentheses. Here, its the exact same process.

Example 6.5.2

Find the following product: $4x^3 (5x^2 - 2x + 5).$

$\begin{array}{rrrrrrl} &5x^2&-&2x&+&5& \\ \times&4x^3&&&&&\text{Multiply} \\ \midrule &20x^5&-&8x^4&+&20x^3&\text{Solution} \end{array}$

Example 6.5.3

Find the following product: $(4x + 7y)(3x - 2y).$

$\begin{array}{rrrrrrl} &4x&+&7y&&& \\ \times &3x&-&2y&&&\text{Multiply} \\ \midrule &12x^2&+&21xy&&&\text{Product of }3x(4x+7y) \\ &&-&8xy&-&14y^2&\text{Product of }-2y(4x+7y) \\ \midrule &12x^2&+&13xy&-&14y^2&\text{Solution} \end{array}$

Example 6.5.4

Find the following product: $(3x^2 + 2x - 5)(4x^2 - 7x + 3).$

$\begin{array}{rrrrrrrrrrl} &3x^2&+&2x&-&5&&&&& \\ \times&4x^2&-&7x&+&3&&&&& \\ \midrule &12x^4&+&8x^3&-&20x^2&&&&&\text{Product of }4x^2(3x^2+2x-5) \\ &&-&21x^3&-&14x^2&+&35x&&&\text{Product of }-7x(3x^2+2x-5) \\ &&&&&9x^2&+&6x&-&15&\text{Product of }3(3x^2+2x-5) \\ \midrule &12x^4&-&13x^3&-&25x^2&+&41x&-&15&\text{Solution} \end{array}$

Example 6.5.5

Find the following product: $(x^3 + 2x^2 + x - 3)(x^3 - 3x^2 - 4x + 6).$

$\begin{array}{rrrrrrrrrrrrrr} &x^3&+&2x^2&+&x&-&3&&&&&& \\ \times &x^3&-&3x^2&-&4x&+&6&&&&&& \\ \midrule &x^6&-&3x^5&-&4x^4&+&6x^3&&&&&& \\ &&&2x^5&-&6x^4&-&8x^3&+&12x^2&&&& \\ &&&&&x^4&-&3x^3&-&4x^2&+&6x&& \\ &&&&&&-&3x^3&+&9x^2&+&12x&+&18 \\ \midrule &x^6&-&x^5&-&9x^4&-&8x^3&+&17x^2&+&18x&+&18 \end{array}$

As seen in the last two examples, the strategy used is that of foiling, with the only difference being that the answers are organized into columns. This eliminates the need to chase terms scattered all over the page, as they are now grouped.

This is the superior strategy to use when multiplying polynomials.

6.6 Special Products

There are a few shortcuts available when multiplying polynomials. When recognized, they help arrive at the solution much quicker.

The first is called a difference of squares. A difference of squares is easily recognized because the numbers and variables in its two factors are exactly the same, but the sign in each factor is different (one plus sign, one minus sign). To illustrate this, consider the following example.

Example 6.6.1

Multiply the following pair of binomials: $(a+b)(a-b).$

$\begin{array}{rrrrr} a&+&b&& \\ a&-&b&& \\ \midrule a^2&+&ab&& \\ &-&ab&-&b^2 \\ \midrule a^2&-&b^2&& \end{array}$

Notice the middle term cancels out and $(a+b)(a-b) = a^2 - b^2$ . Cancelling the middle term during multiplication is the same for any difference of squares.

Example 6.6.2

Multiply the following pair of binomials: $(x - 5)(x + 5).$

Recognize a difference of squares. The solution is $x^2 - 25.$

Example 6.6.3

Multiply the following pair of binomials: $(3x + 7)(3x - 7).$

Recognize a difference of squares. The solution is $9x^2 - 49.$

Example 6.6.4

Multiply the following pair of binomials: $(2x - 6y)(2x + 6y).$

Recognize a difference of squares. The solution is $4x^2 - 36y^2.$

Another pair of binomial multiplications useful to know are perfect squares. These have the form of $(a + b)^2$ or $(a - b)^2$ .

Example 6.6.5

Multiply the following pair of binomials: $(a + b)^2$ and $(a - b)^2.$

$\begin{array}{rr} \begin{array}{rrrrrr} &(a&+&b)&& \\ \times &(a&+&b)&& \\ \midrule &a^2&+&ab&& \\ &&+&ab&+&b^2 \\ \midrule &a^2&+&2ab&+&b^2 \end{array}\hspace{0.5in} & \begin{array}{rrrrrr} &(a&-&b)&& \\ \times &(a&-&b)&& \\ \midrule &a^2&-&ab&& \\ &&-&ab&+&b^2 \\ \midrule &a^2&-&2ab&+&b^2 \end{array} \end{array}$

The pattern of multiplication for any perfect square is the same. The first term in the answer is the square of the first term in the problem. The middle term is 2 times the first term times the second term. The last term is the square of the last term.

$(a + b)^2 = a^2 + 2ab + b^2 \text{ and }(a - b)^2 = a^2 - 2ab + b^2$