7. Answer: P(x) = [tex]\bold{\dfrac{3}{4}}[/tex](x³ 1 x² - 10x + 6)
Step-by-step explanation:
The factored form of a polynomial is: a(x - p)(x-q)...(x-z) . where p, q, ..., z are the roots.
Convert the roots into factors, include conjugates of any complex roots, and multiply all of the factors. Then find the a-value.
roots: x = 3 x = 2 + √2 x = 2 - √2 (conjugate)
factors: x - 3 = 0 x - 2 - √2 = 0 x - 2 + √2 = 0
(x - 3)(x - 2 - √2)(x - 2 + √2)
multiply the complex roots:
(x + 3)(x² - 4x + 2)
distribute (x + 3) to find the basic polynomial
x³ + x² - 10x + 6
Next, find the a-value:
P(x) = a(x³ + x² - 10x + 6)
P(1) = a(1 - 1 - 10 + 6) = -3
-4a = -3
[tex]a=\dfrac{3}{4}[/tex]
[tex]\boxed{P(x)=\dfrac{3}{4}(x^3-x^2-10x+6)}[/tex]
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8. Answer: P(x) = [tex]\bold{\dfrac{1}{2}}[/tex](x⁴ + 8)
Step-by-step explanation:
The factored form of a polynomial is: a(x - p)(x-q)...(x-z) . where p, q, ..., z are the roots.
Convert the roots into factors, include conjugates of any complex roots, and multiply all of the factors. Then find the a-value.
roots: x = -2 x = 1 + i x = 1 - i (conjugate)
factors: x + 2 = 0 x - 1 - i = 0 x - 1 + i = 0
The polynomial has a degree of 4, so x = -2 must be a double root.
(x + 2)²(x - 1 - i)(x - 1 + i)
multiply the double root & complex roots:
(x² + 4x + 4)(x² - 2x + 2)
distribute (x² + 4x + 4) to find the basic polynomial
x⁴ + 8
Next, find the a-value:
P(x) = a(x⁴ + 8)
P(0) = a(0 + 8) = 4
8a = 4
[tex]a=\dfrac{1}{2}[/tex]
[tex]\boxed{P(x)=\dfrac{1}{2}(x^4+8)}[/tex]
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9. Answer: P(x) = [tex]\bold{-\dfrac{1}{2}}[/tex](x⁴ - x³ - 6x²)
Step-by-step explanation:
The factored form of a polynomial is: a(x - p)(x-q)...(x-z) . where p, q, ..., z are the roots. The roots are where the graph crosses the x-axis.
roots: x = -2 x = 0 x = 3
factors: x + 2 = 0 x = 0 x - 3 = 0
The graph "touches - not crosses" the x-axis at x = 0 so x = 0 is a double root.
(x + 2)(x)²(x - 3)
multiply the factors (x + 2)(x - 3):
x²(x² - x - 6)
distribute (x)² to find the basic polynomial
x⁴ - x³ - 6x²
Next, find the a-value:
P(x) = a(x⁴ - x³ - 6x²)
P(2) = a[(2)⁴ -(2)³ - 6(2)²] = 8
a(16 - 8 - 24) = 8
-16a = 8
[tex]a=-\dfrac{1}{2}[/tex]
[tex]\boxed{P(x)=-\dfrac{1}{2}(x^4-x^3-6x^2)}[/tex]
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10. Answer: P(x) = -(x³ - 2x² + x - 2)
Step-by-step explanation:
The graph displayed is an x³ graph that has been shifted to the left 2 units and reflected over the x-axis. Only one root is shown, which means the other two roots are complex numbers. Since P(i) = 0, then another root is: x = i
roots: x = -2 x = i x = - i (conjugate)
factors: x + 2 = 0 x - i = 0 x + i = 0
(x + 2)(x - i)(x + i)
multiply the complex roots:
(x + 2)(x² + 1)
distribute (x + 2) to find the basic polynomial
x³ - 2x² + x - 2
Next, reflect over the x-axis:
P(x) = - (x³ - 2x² + x - 2)
