This document provides an overview of polynomials, including adding, subtracting, and multiplying polynomials. It begins by reviewing integer rules and exponent rules. It then covers combining like terms, adding polynomials by combining like terms and using integer addition/subtraction rules, subtracting polynomials using distribution and combining like terms, and multiplying polynomials using distribution and the box method. Examples are provided for each topic.

1. Unit 3
Polynomials
Adding Polynomials
Subtracting Polynomials
Multiplying Polynomials
By Genny Simpson

2. Let’s Review!
Integer rules
Exponent rules
Combining like-terms

3. Before you can add or subtract polynomials you need to review the
integer rules.
To add integers, add if they have the same sign and keep the sign.
For instance -4 + -6 = -10 and 5 + 12 = 17.
Subtract if they have different signs and take the sign of the larger
number. For instance 10 + -6 = 4 and -10 + 6 = -4.
To subtract integers, change the subtraction sign to adding the
opposite and then follow the rules for addition. For instance -2 – 3 =
-2 + -3 = -5 and 6 – (-5) = 6 + 5 = 11.
INTEGER RULES

4. The only rule we really need for now is the multiplication rule.
W hen you multiply bases, you multiply the numbers in front, the
coefficients, and add the exponents.
For example: 3x · 2x = 6x2 and 4x2 · 5x = 20x3.
EXPONENT RULES

5. To combine like terms they must be the same except for the
numbers in front, the coefficients.
Here are some examples of like terms: 3x and -5x, 2y3 and 7y3,
-2x2 and 9x2.
To combine 3x and -5x, you use the integer rule for adding different
signs. Subtract 3 from 5 and keep the negative or -2x.
To combine 2y3 and 7y3, use the addition rule for integers: same
signs, add and keep the sign. So 2y3 + 7y3 = 9y3.
To combine -2x2 and 9x2, you use the addition rule again, subtract
and keep the sign of the larger number. So -2x2 + 9x2 = 7x2.
COMBINING LIKE TERMS

6. To add polynomials, we are using the integer rules for addition and
subtraction and combining like terms.
For example: (2x2 – 3x + 5) + (4x2 + 5x – 6) Put like terms together
and combine them. (2x2 + 4x2) + (-3x + 5x) + (5 - 6 ) = 6x2 + 2x – 1.
Here’s another example: (5d – 2d2 + 6) + (5d2 + 2). Rearrange to
keep like terms together. Be sure to bring the sign in front of the
term with it when you rearrange. (-2d2 + 5d2) + 5d + (6 + 2).
So the answer is 3d2 + 5d + 8.
ADDING POLYNOMIALS

7. To subtract polynomials, we use the integer rules for addition and
subtraction and combining like terms as well as the distributive
property.
For example: (6c2 – 2c – 7) – (3c2 – 6c + 1). Before we rearrange
to put like terms together, we need to distribute the negative
through the second parentheses. (6c2 – 2c – 7) + (-3c2 + 6c - 1).
Now we can rearrange and group like terms.
(6c2 + -3c2) + (-2c + 6c) + (-7 – 1) = 3c2 + 4c – 8.
SUBTRACTING POLYNOMIALS

8. There are two methods we can use to multiply
polynomials, the distributive property and the box
method.
Let’s look first at the distributive property. First distribute by x.
(x + 2)(x2 – 3x + 4) = x · x2 + x · -3x + x · 4 = x3 – 3x2 + 4x
Then distribute by 2. 2 · x2 + 2 · -3x + 2 · 4 = 2x2 – 6x + 8.
Now all we do is combine like terms: x3 – x2 – 2x + 8.
MULTIPLYING
POLYNOMIALS

9. Now let’s take a look at the Box Method!
To multiply (2x + 3)(x2 – 2x + 7) we need to draw a box with 2 rows
and three columns. We label the rows with terms of the binomial
and the columns with the terms of the trinomial. It looks like this:
x2 -2x +7
2x 2x3 -4x2 14x
+3 3x2 -6x 21
Notice that the like terms are on the diagonals.
So the answer is 2x3 – x2 + 8x + 21.
MORE MULTIPLICATION

10. Now it’s your turn! Practice adding, subtracting,
and multiplying polynomials.
1. (4x3 – 2x2 + 5) + (-10x2 + 6x + 7)
2. (-15x2 + 7x – 11) – (-11x2 – 13x + 9)
3. (16x – 13) – (3x2 + 5x – 10) + (7x – 12x2 + 8)
4. 5x3(2x2 – 4x + 15)
5. (3x – 7)(2x + 9)
6. (6x + 5)(3x2 – x – 4)

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