Lesson #11 Polynomials
A polynomial is a variable expression with one (or more) variable appearing to various positive integer exponents. For example, 3y2 + 2y + 9 and 4x10 - 3x5 + 2x2 - 5 are polynomials. The prefix poly means "several", and the root nomial refers to the individual terms. So polynomial simply means "several terms". Remember that the terms are the separate multiplicative parts in an algebraic expression. Each term starts with a + or - sign, and continues through the number multiplier (coefficient), variable(s) and exponent(s). The next + or - sign starts the next term. The degree of a polynomial is the largest exponent appearing in any one term. For example, the polynomial 3y2 + 2y + 9 has three terms (3y2 and +2y and +9) and degree 2 (it uses y2). The polynomial 4x10 - 3x5 + 2x2 - 5 has 4 terms (4x10 and - 3x5 and + 2x2 and - 5) and degree 10. The degree is determined only by the highest exponent used. None of the lower exponent terms affect the degree.
Evaluate polynomials just as any other evaluation problem: replace each occurrence of the variable with the given value, and then simplify according to the Order of Operations.
Example 1. Evaluate 5x2 - 4x + 1 when x = -2.
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5x2 - 4x + 1 |
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= 5(-2)2 - 4(-2) + 1 |
Replace the variable with the given value (-2) |
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= 5(4) -4(-2) + 1 |
Follow the Order of Operations to simplify |
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= 20 + 8 + 1 |
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= 29 |
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Example 2. Evaluate a3 - 4a when a = 5.
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a3 - 4a |
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= (5)3 - 4(5) |
Replace each letter a by (5) |
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= 125 - 4(5) |
Do the exponent first, followed by the multiplication |
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= 125 - 20 |
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= 105 |
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Add or subtract polynomials by combining only the like terms. Like terms are those with the same variable part, including the same exponent. Note that the exponents are never changed in an addition or subtraction problem; only the coefficients of like terms are added, with the variable and its exponent left alone.
Example 3. Simplify (3x2 - 7x + 2) + (4x2 + 5x - 9).
This expression has three distinct type of terms: those with x2, those with x, and those with no x. Add each type separately.
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(3x2 - 7x + 2) + (4x2 + 5x - 9) |
With a + sign only between parentheses, drop the parentheses and group like variable terms. Add coefficients for each type of term. |
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= (3x2 + 4x2) + (-7x + 5x) + (2 - 9) |
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= 7x2 - 2x - 7 |
Example 4. Simplify (2y2 + 5y - 3) - (5y2 - 6y + 1).
To subtract a polynomial from another, first remove the parentheses by multiplying each term by -1. This has the effect of changing each sign inside the following parentheses.
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(2y2 + 5y - 3) - (5y2 - 6y + 1) |
With a - sign between parentheses, change each sign in the following parentheses, and group like variable terms. Add coefficients for each type of term. |
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= 2y2 + 5y - 3 - 5y2 + 6y - 1 |
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= (2y2 - 5y2) + (5y + 6y) + (-3 - 1) |
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= -3y2 + 11y - 4 |
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Multiply polynomials by multiplying to each term inside the next parentheses. For each term, multiply coefficients and add exponents.
Example 5. Simplify 3x(5x - 2).
This expression asks you to multiply 3x by each of 5x and -2.
That is, 3x(5x - 2) = 3x× 5x and 3x(-2) = 15x2 - 6x. Note that the first term uses x× x = x2, while the second term has only x. This answer cannot be simplified further because the two terms have different variable parts.
Example 6. Simplify -2x2(3x2 + 4x - 5).
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-2x2(3x2 + 4x - 5) |
= -2x2× 3x2 - 2x2× 4x - 2x2× (-5) |
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= -6x4 - 8x3 + 10x2 |
Be especially careful with the sign on each product. Add the exponents to get the correct total power for each variable term, but do not try to combine terms with different variable parts.
Example 7. Simplify (3y - 2)(5y - 2).
To multiply one polynomial by another, you will need to multiply each term in the first by each term in the second. The FOIL method is a common device for remembering the required products in a two-term by two-term multiplication: First, Outer, Inner, Last.
The product of the First terms is 3y× 5y = 15y2.
The product of the Outer terms is 3y× (-2) = -6y.
The product of the Inner terms is -2× 5y = -10y.
The product of the Last terms is -2× (-2) = +4.
Put these all together to get (3y - 2)(5y - 2) = 15y2 - 6y - 10y + 4 = 15y2 - 16y + 4
(add the two like terms in the middle in the last step).
Exercises
For each polynomial, list the degree and the number of terms.
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1. 5x3 - 3x + 7 |
2. 5b3 - 7b2 - b + 1 |
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3. 12x8 - 4x4 |
4. 6y12 - 2y7 - 3y2 + 91 |
Evaluate:
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5. 3x2 + 2x + 9 when x = 2 |
6. 4y3 - y + 8 when y = -1 |
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7. 12x9 + x7 - x6 + 3x2 + 17 when x = 0 |
8. b2 - 9 when b = -4 |
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9. 3x3 - 4x + 2 when x = 1 |
10. x2 - y2 + 4x when x = -2 and y = 3 |
Simplify each of the following.
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11. (4x + 1) + (3x - 5) |
12. (x2 + 2x - 8) + (3x2 + 3) |
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13. (a2 - 4a + 9) - (5a2 - 7a + 6) |
14. (2x3 + 3x) - (4x2 + 2x + 1) |
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15. (7b + 3) + (9b8 - 2b + 5) |
16. (3x2 - 5x + 1) - (x2 + 2x - 4) |
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17. 2x(9x - 4) |
18. (w2 - w - 1) - (w2 - 3w - 4) |
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19. 3y3(y2 - 3y + 8) |
20. -q2(2q + 5) |
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21. x2(2x + 1) |
22. 3x(4x + 5) |
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23. (-2x)(x2 + 2x + 1) |
24. (a + 3)(5a2) |
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25. (x + 3)(5x2 + 1) |
26. (2x2 + 1)(4x - 3) |
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27. (y + 2)(y - 3) |
28. (2z + 3)(z + 8) |
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29. (5x - 1)(x - 2) |
30. (6x + 2) + (3x - 9) |
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31. (q2 - 3q + 4) - (3q2 - 9) |
32. (2t - 9)(t - 1) |
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33. (5y3 - y + 8) - (4y - 3) |
34. (w - 3)(w - 2) |
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35. (x + 1)(x2 + x + 2) |
36. (x - 2)(x2 + 3x - 5) |
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37. (2c - 1)(c2 -2c - 5) |
38. (x2 + 2x + 1)(x2 - 3x + 4) |
Check the answers to Lesson 11 exercises
Go on to Lesson 12