Polynomials
Polynomials are expressions of more than two algebraic terms. The sum of several terms that contain different powers of the same variable(s). The sum of 1 or more terms each of which is the product of a collection of numbers and letters. It can have any term having constants, variable, and exponents. The exponents should be the whole number.
Answers:
- D.
- C.
- A.
- B.
- A.
- D.
- A.
- A.
- D.
- D.
- D.
Solutions:
1. D. is a binomial.
2. x² - 25 is a difference of two squares whose factors are (x - 5)(x + 5).
3. 5x¹⁰ + 4x¹² + 4x⁶ + x⁴ – x. The leading coefficient is 4 because the highest exponent is 12.
4. 3x⁴ + 5x² – 2x – 4 divided by x + 4 in synthetic division is
-4⊥ 3 0 5 -2 -4 since there is no term of the third degree so we will place 0.
5. f(2) = 2(2)³ – 5(2)² + 6(2) – 11
f(2) = 2(8) - 5(4) + 12 - 11
f(2) = 16 - 20 + 12 - 11
f(2) = 16 + 12 - 20 - 11
f(2) = 28 - 31
f(2) = -3
6. 2 is a zero of f(x)
7. P(x) = 5x³ – 2x² + 6x – 8
Factors of 5 = ±1, ±5
Factors of -8 = ±1, ±2, ±4, ±8
Therefore, there are 12 possible rational zeros.
8. 3x³ – 4x² + 3x – 5 = 0
if x = [tex]\frac{3}{5}[/tex]
3([tex]\frac{3}{5}[/tex])³ – 4([tex]\frac{3}{5}[/tex])² + 3([tex]\frac{3}{5}[/tex]) – 5 = 0
3([tex]\frac{27}{125}[/tex]) - 4([tex]\frac{9}{25}[/tex]) + [tex]\frac{9}{5}[/tex] - 5 = 0
[tex]\frac{81}{125}[/tex] - [tex]\frac{36}{25}[/tex] + [tex]\frac{9}{5}[/tex] - 5 = 0
[tex]\frac{81 - 180 + 225 - 625}{125}[/tex] = 0
[tex]\frac{81 + 225 - 180 - 625}{125}[/tex] = 0
[tex]\frac{306 - 805}{125}[/tex] = 0
[tex]\frac{-499}{125}[/tex] = 0
-3 [tex]\frac{124}{125}[/tex] ≠ 0
9. x(x + 3)(x + 3)(x – 1)(2x + 1) = 0
The roots are: x = 0, x = -3, x = 1, x = -[tex]\frac{1}{2}[/tex] so the answer is D.
10. x²(x + 1)³ (x – 3) (x + 4)² = 0
11. x²(x + 1)³ (x – 3) (x + 4)² = 0
2 + 3 + 1 + 2 = 8
Therefore, the polynomial is in the 8th degree.
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