3. If a polynomial equation p(x) = 0 has 3 + 4i as a solution, what other solutions must it have?
2 answers:
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Part (i)
<h3>Answer: x^2 + 5x + 6</h3>-----------------
Work Shown:
(x+3)(x+2)
y(x+2) ..... Let y = x+3
y*x + y*2 ... distribute
x(y) + 2(y)
x(x+3) + 2(x+3) .... plug in y = x+3
x*x + x*3 + 2*x + 2*3 ... distribute
x^2 + 3x + 2x + 6
x^2 + 5x + 6
=====================================================
Part (ii)
<h3>Answer: 4x^2 - 16x + 7</h3>-----------------
Work Shown:
We could follow the same set of steps as shown back in part (i), but I'll show a different approach. Feel free to use the method I used back in part (i) if the visual approach doesn't make sense.
The diagram below is a visual way to organize all the terms. Many textbooks refer to it as "the box method" which helps multiply out any two algebraic expressions.
Each inner cell is found by multiplying the corresponding outer terms. For instance, in the upper left corner we have 2x*2x = 4x^2. The other cells are filled out the same way.
The terms in those four inner cells (gray boxes) are:
- 4x^2
- -14x
- -2x
- 7
The like terms here are -14x and -2x which combine to -16x, since -14+(-2) = -16.
We end up with the answer 4x^2-16x+7
