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- 1. Unit 3 PolynomialsAdding PolynomialsSubtracting PolynomialsMultiplying PolynomialsBy Genny Simpson
- 2. Let’s Review!Integer rulesExponent rulesCombining like-terms
- 3. Before you can add or subtract polynomials you need to review theinteger 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 largernumber. For instance 10 + -6 = 4 and -10 + 6 = -4.To subtract integers, change the subtraction sign to adding theopposite 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, thecoefficients, 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 thenumbers 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 differentsigns. Subtract 3 from 5 and keep the negative or -2x.To combine 2y3 and 7y3, use the addition rule for integers: samesigns, add and keep the sign. So 2y3 + 7y3 = 9y3.To combine -2x2 and 9x2, you use the addition rule again, subtractand 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 andsubtraction and combining like terms.For example: (2x2 – 3x + 5) + (4x2 + 5x – 6) Put like terms togetherand combine them. (2x2 + 4x2) + (-3x + 5x) + (5 - 6 ) = 6x2 + 2x – 1.Here’s another example: (5d – 2d2 + 6) + (5d2 + 2). Rearrange tokeep like terms together. Be sure to bring the sign in front of theterm 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 andsubtraction and combining like terms as well as the distributiveproperty.For example: (6c2 – 2c – 7) – (3c2 – 6c + 1). Before we rearrangeto put like terms together, we need to distribute the negativethrough 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 multiplypolynomials, the distributive property and the boxmethod.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 + 4xThen distribute by 2. 2 · x2 + 2 · -3x + 2 · 4 = 2x2 – 6x + 8.Now all we do is combine like terms: x3 – x2 – 2x + 8.MULTIPLYINGPOLYNOMIALS
- 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 rowsand three columns. We label the rows with terms of the binomialand 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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