Operations on polynomials

Examples of like terms: $8x^2$ and $-4x^2$; $-2xy$ and $11xy$; $3xy^3$ and $xy^3$

Example 1: Simplify, $(-2x^2+3y-y^2)-(4x^2+2y-5y^2)$

Solution:

First remove the parentheses and then combine the like terms:

$\begin{align*} (-2x^2+3y-y^2)-(4x^2+2y-5y^2)&=-2x^2+3y-y^2-4x^2-2y+5y^2\\ &=-6x^2+y+4y^2 \end{align*}$

Example 2: Simplify, $(3a^2b-4ab+4ab^2)-(5a^2b-8ab+ab^2)$

Solution:

$\begin{align*} (3a^2b-4ab+4ab^2)-(5a^2b-8ab+ab^2)&=3a^2b-4ab+4ab^2-5a^2b+8ab-ab^2\\ &=-2a^2b+4ab+3ab^2 \end{align*}$

Example 1: Multiply and simplify, $(4x-2)(3x+5)$

Solution:

$\begin{align*} (4x-2)(3x+5)&=4x\cdot 3x+4x\cdot 5-2\cdot 3x-2\cdot5\\ &= 12x^2+20x-6x-10\\ &=12x^2+14x-10 \end{align*}$

Example 2: Write without parenthesis and simplify, $(4x-3u)^2$

Solution:

$\begin{align*} (4x-3u)^2 &=(4x-3u)(4x-3u)\\ &= 16x^2-12xu-12xu+9u^2\\ &=16x^2-24xu+9u^2 \end{align*}$

Example 3: Multiply and simplify, $(-4y-3)(5y^2-2y+5)$

Solution:

$\begin{align*} (-4y-3)(5y^2-2y+5) &=-20y^3+8y^2-20y-15y^2+6y-15\\ &= -20y^3-7y^2-14y-15 \end{align*}$

Example: Divide, $\dfrac{24x^4y-8x^3y+12x^2y^5}{4x^2y}$. Simplify the answer as much as possible.

Solution:

$\begin{align*} \dfrac{24x^4y-8x^3y+12x^2y^5}{4x^2y} &= \dfrac{24x^4y}{4x^2y}-\dfrac{8x^3y}{4x^2y}+\dfrac{12x^2y^5}{4x^2y} \\ &= 6x^2-2x+3y^4 \end{align*}$

Example: Divide, $12x^3+29x^2+11x-30 \div 4x+3$

Solution:

We have the dividend:

$12x^3+29x^2+11x-30 $

and the divisor:

$4x+3$

1. Arrange the dividend and the divisor in descending order — that is, in descending powers of the variable.

$12x^3+29x^2+11x-30 $

2. If a term is missing — for example, you have an $x^3$ term and an $x$ term but no $x^2$ term — then add an $x^2$ term with a zero coefficient between the $x^3$ and $x$ terms.

Next, start the division,

$4x+3\overline{)12x^3+29x^2+11x-30}$

Now, take the leading term of the dividend and divide that by the leading term of the divisor.

$\dfrac{12x^3}{4x}=3x^2$. Put this ratio in the quotient.

$ \hspace{15.5mm}3x^2$
$ 4x+3\overline{)12x^3+29x^2+11x-30}$

Next, multiply the ratio by the divisor:

$3x^2(4x+3)=12x^3+9x^2$. Write this below and subtract,

$ \hspace{15.5mm}3x^2$
$ 4x+3\overline{)12x^3+29x^2+11x-30}$
$\hspace{16mm}\underline{12x^3+9x^2}$
$\hspace{31.5mm}20x^2$

Bring the next term (the $x$ term) down

$ \hspace{15.5mm}3x^2$
$ 4x+3\overline{)12x^3+29x^2+11x-30}$
$\hspace{16mm}\underline{12x^3+9x^2}$
$\hspace{31.5mm}20x^2+11x$

Take the ratio of the leading term of the last expression to the leading term of the divisor.

$\dfrac{20x^2}{4x}=5x$. Add this to the quotient.

$ \hspace{15.5mm}3x^2+5x$
$ 4x+3\overline{)12x^3+29x^2+11x-30}$
$\hspace{16mm}\underline{12x^3+9x^2}$
$\hspace{31.5mm}20x^2+11x$

Multiply the ratio by the divisor:

$5x(4x+3)=20x^2+15x$. Now subtract this,

$ \hspace{15.5mm}3x^2+5x$
$ 4x+3\overline{)12x^3+29x^2+11x-30}$
$\hspace{16mm}\underline{12x^3+9x^2}$
$\hspace{31.5mm}20x^2+11x$
$\hspace{31.5mm}\underline{20x^2+15x}$
$\hspace{43.5mm}-4x$

Bring the next term, $-30$ down

$ \hspace{15.5mm}3x^2+5x$
$ 4x+3\overline{)12x^3+29x^2+11x-30}$
$\hspace{16mm}\underline{12x^3+9x^2}$
$\hspace{31.5mm}20x^2+11x$
$\hspace{31.5mm}\underline{20x^2+15x}$
$ \hspace{43.5mm}-4x-30$

Divide the leading term by the leading term of the divisor and add it to the quotient.

$\dfrac{-4x}{4x}=-1$

Then, multiply this ratio by the divisor and subtract as before.

$-1(4x+3)=-4x-3$,

$ \hspace{15.5mm}3x^2+5x-1$
$ 4x+3\overline{)12x^3+29x^2+11x-30}$
$\hspace{16mm}\underline{12x^3+9x^2}$
$\hspace{31.5mm}20x^2+11x$
$\hspace{31.5mm}\underline{20x^2+15x}$
$ \hspace{43.5mm}-4x-30$
$ \hspace{45.5mm}\underline{-4x-3}$
$ \hspace{55.5mm}-27$

Now, write the answer,

Quotient: $3x^2+5x-1$

Remainder: $-27$