learning task 2 a determine the factors of the given polynomials 1 x3 plus 2x2 - 5x - 6 2. x3 plus x2 -x-1 3.x3 - x2 -10x -8
Answers
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Factoring Polynomials
Factor theorem, a theorem linking factors and zeros of polynomials. Special case of the polynomial remainder theorem. Is often used to factor and find the roots of polynomials. Root or zero is where the polynomial is equal to zero. It simply states that when f(k) = 0, then (x – k) is a factor of f(x).
Learning Task 2:
Answers:
A.
(x + 1)(x - 2)(x + 3)
(x + 1)(x + 1)(x - 1)
(x+ 1)(x - 2)(x - 4)
B.
1.
a. 3x - 1
b. when divided into 2 the length will be and width
2. 2x³ -3x² + 4x - 6
Solutions:
A.
1. x³ + 2x² - 5x - 6
Factor the first term.
x³ = (x)(x)(x)
Factor the last term.
-6 = (-1)(6), (-2)(3), (-3)(2), (-6)(1)
Using the factor theorem, check which among the factors of -6 is the correct factor.
if x = -1 then
x³ + 2x² - 5x - 6 = 0
(-1)³ + 2(-1)² - 5(-1) - 6 = 0
-1 + 2(1) + 5 - 6 = 0
-1 - 6 + 2 + 5 = 0
-7 + 7 = 0
0 = 0
Therefore, x + 1 is a factor.
Find other factors.
if x = 2
(2)³ + 2(2)² - 5(2) - 6 = 0
8 + 2(4) - 10 - 6 = 0
8 + 8 - 10 - 6 = 0
16 - 16 = 0
0 = 0
Therefore, x - 2 is another factor.
Find the last factor.
if x = -3
(-3)³ + 2(-3)² - 5(-3) - 6 = 0
-27 + 2(9) + 15 - 6 = 0
-27 + 18 + 15 - 6 = 0
-33 + 33 = 0
0 = 0
Therefore, x + 3 is the last factor.
2. x³ + x² - x - 1
Factor the first term.
x³ = (x)(x)(x)
Factor the last term.
-1 = (1)(-1)
Using the factor theorem, let us check if x + 1 and x - 1 is a factor.
if x = -1
(-1)³ + (-1)² - (-1) - 1 = 0
-1 + 1 + 1 - 1 = 0
-2 + 2 = 0
0 = 0
Therefore, x + 1 is a factor.
if x = 1
(1)³ + (1)² - (1) - 1 = 0
1 + 1 - 1 - 1 = 0
2 - 2 = 0
0 = 0
Therefore x - 1 is also a factor.
Find the third factor.
Since we already have (x + 1) and (x - 1) which give s a product of (x² - 1), the last we are looking for will give a product of x³ for the first term and -1 for the last term. Thus, we need (x + 1).
Therefore, the factors are (x + 1)(x + 1) and (x - 1).
3. x³ - x² - 10x - 8
Factor the first term.
x³ = (x)(x)(x)
Factor the last term.
-8 = (-1)(8), (-2)(4), (-8)(1), (-4)(2)
Using the factor theorem, let us check which are the factors of the given polynomial.
if x = -1
(-1)³ - (-1)² - 10(-1) - 8 = 0
-1 - 1 + 10 - 8 = 0
-10 + 10 = 0
0 = 0
Therefore, x + 1 is a factor of the polynomial.
if x = -2
(-2)³ - (-2)² - 10(-2) - 8 = 0
-8 - 4 + 20 - 8 = 0
-20 + 20 = 0
0 = 0
Therefore, (x + 2) is also a factor.
Find the last factor.
if x = 4
(4)³ - (4)² - 10(4) - 8 = 0
64 - 16 - 40 - 8 = 0
64 - 64 = 0
0 = 0
Therefore, (x - 4) is the last factor.
B.
1.
a. Given: Area = 3x² + 5x - 6
width = x - 2
Find the length.
Area = length x width
3x² + 5x - 6 = (length)(x - 2)
= length
3x - 1
x - 2
3x² - 6x
-x - 2
x + 2
0
Therefore, the length of the garden is 3x - 1
b. 3x² + 5x - 2 ÷ 2 = (3x - 1)(x + 2)
2
2. Given: 3x - 4 - costs of one ream of bond paper
6x⁴ - 17x³ + 24x² - 34x + 24 - total amount spent for reams of bond papers
Find the number of reams of bond papers bought.
2x³ -3x² + 4x - 6
3x - 4 √6x⁴ - 17x³ + 24x² - 34x + 24
6x⁴ - 8x³
- 9x³ + 24x² - 34x + 24
- 9x³ + 12x²
12x² - 34x + 24
12x² - 16x
- 18x + 24
- 18x + 24
0
Therefore, there are 2x³ -3x² + 4x - 6 reams of bond papers bought
Step-by-step explanation: