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]
