Find trusted answers to your questions with the help of IDNLearn.com's knowledgeable community. Join our interactive Q&A platform to receive prompt and accurate responses from experienced professionals in various fields.
Sagot :
Certainly! Let's break down the solution to each part of the question step-by-step:
### (a) Use the fundamental theorem of algebra to determine the number of roots
The fundamental theorem of algebra states that every non-zero polynomial of degree [tex]\( n \)[/tex] with complex coefficients has exactly [tex]\( n \)[/tex] roots in the complex number system (counting multiplicities).
Consider the polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex]:
- The highest exponent of [tex]\( x \)[/tex] in this polynomial is 2.
- Therefore, the polynomial is of degree 2.
According to the fundamental theorem of algebra, a polynomial of degree 2 has exactly 2 roots in the complex number system.
### (b) What are the roots of [tex]\( 2x^2 + 4x + 7 \)[/tex]?
To find the roots of the polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex], we need to solve the equation [tex]\( 2x^2 + 4x + 7 = 0 \)[/tex].
Step 1: Identify the coefficients.
The equation is of the form [tex]\( ax^2 + bx + c = 0 \)[/tex] where:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 7 \)[/tex]
Step 2: Use the quadratic formula to find the roots.
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Step 3: Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
[tex]\[ x = \frac{-4 \pm \sqrt{4^2 - 4(2)(7)}}{2(2)} \][/tex]
Step 4: Simplify inside the square root.
[tex]\[ x = \frac{-4 \pm \sqrt{16 - 56}}{4} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{-40}}{4} \][/tex]
Since the discriminant ([tex]\( b^2 - 4ac \)[/tex]) is negative ([tex]\( -40 \)[/tex]), the roots will be complex numbers.
Step 5: Simplify the expression further.
[tex]\[ x = \frac{-4 \pm \sqrt{-40}}{4} \][/tex]
Recall that [tex]\( \sqrt{-1} = i \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit.
Thus,
[tex]\[\sqrt{-40} = \sqrt{40} \cdot i = \sqrt{4 \cdot 10} \cdot i = 2\sqrt{10} \cdot i \][/tex]
Now substitute this back into the expression:
[tex]\[ x = \frac{-4 \pm 2\sqrt{10} \cdot i}{4} \][/tex]
Step 6: Split the fraction.
[tex]\[ x = \frac{-4}{4} \pm \frac{2\sqrt{10} \cdot i}{4} \][/tex]
[tex]\[ x = -1 \pm \frac{\sqrt{10} \cdot i}{2} \][/tex]
Therefore, the roots are:
[tex]\[ x = -1 - \frac{\sqrt{10} \cdot i}{2} \][/tex]
[tex]\[ x = -1 + \frac{\sqrt{10} \cdot i}{2} \][/tex]
### Final Answer:
(a) The polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex] has exactly 2 roots.
(b) The roots of the polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex] are:
[tex]\[ x = -1 - \frac{\sqrt{10} \cdot i}{2} \][/tex]
[tex]\[ x = -1 + \frac{\sqrt{10} \cdot i}{2} \][/tex]
### (a) Use the fundamental theorem of algebra to determine the number of roots
The fundamental theorem of algebra states that every non-zero polynomial of degree [tex]\( n \)[/tex] with complex coefficients has exactly [tex]\( n \)[/tex] roots in the complex number system (counting multiplicities).
Consider the polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex]:
- The highest exponent of [tex]\( x \)[/tex] in this polynomial is 2.
- Therefore, the polynomial is of degree 2.
According to the fundamental theorem of algebra, a polynomial of degree 2 has exactly 2 roots in the complex number system.
### (b) What are the roots of [tex]\( 2x^2 + 4x + 7 \)[/tex]?
To find the roots of the polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex], we need to solve the equation [tex]\( 2x^2 + 4x + 7 = 0 \)[/tex].
Step 1: Identify the coefficients.
The equation is of the form [tex]\( ax^2 + bx + c = 0 \)[/tex] where:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 7 \)[/tex]
Step 2: Use the quadratic formula to find the roots.
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Step 3: Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
[tex]\[ x = \frac{-4 \pm \sqrt{4^2 - 4(2)(7)}}{2(2)} \][/tex]
Step 4: Simplify inside the square root.
[tex]\[ x = \frac{-4 \pm \sqrt{16 - 56}}{4} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{-40}}{4} \][/tex]
Since the discriminant ([tex]\( b^2 - 4ac \)[/tex]) is negative ([tex]\( -40 \)[/tex]), the roots will be complex numbers.
Step 5: Simplify the expression further.
[tex]\[ x = \frac{-4 \pm \sqrt{-40}}{4} \][/tex]
Recall that [tex]\( \sqrt{-1} = i \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit.
Thus,
[tex]\[\sqrt{-40} = \sqrt{40} \cdot i = \sqrt{4 \cdot 10} \cdot i = 2\sqrt{10} \cdot i \][/tex]
Now substitute this back into the expression:
[tex]\[ x = \frac{-4 \pm 2\sqrt{10} \cdot i}{4} \][/tex]
Step 6: Split the fraction.
[tex]\[ x = \frac{-4}{4} \pm \frac{2\sqrt{10} \cdot i}{4} \][/tex]
[tex]\[ x = -1 \pm \frac{\sqrt{10} \cdot i}{2} \][/tex]
Therefore, the roots are:
[tex]\[ x = -1 - \frac{\sqrt{10} \cdot i}{2} \][/tex]
[tex]\[ x = -1 + \frac{\sqrt{10} \cdot i}{2} \][/tex]
### Final Answer:
(a) The polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex] has exactly 2 roots.
(b) The roots of the polynomial [tex]\( 2x^2 + 4x + 7 \)[/tex] are:
[tex]\[ x = -1 - \frac{\sqrt{10} \cdot i}{2} \][/tex]
[tex]\[ x = -1 + \frac{\sqrt{10} \cdot i}{2} \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.