Complex Numbers
Operations on Complex Numbers
Grade 11
Question:
<p>If <span class="inline-math">x = 4 + 3i\</span> (where <span class="inline-math">i = \sqrt{-1}\</span>), then the value of <span class="inline-math">x^3 - 4x^2 - 7x + 12\</span> equals:</p>
<p>(a) -88</p>
<p>(b) <span class="inline-math">48 + 36i\</span></p>
<p>(c) <span class="inline-math">-256 + 12i\</span></p>
<p>(d) -84</p>
Step-by-Step Solution
Key Concept: Substitute the complex number x = 4 + 3i into the polynomial expression x³ - 4x² - 7x + 12 and carefully compute powers of the complex number using (a + bi)² = a² - b² + 2abi, then combine like terms to get a real result.
Step 1: Calculate $x^2$
Given $x = 4 + 3i$.
$$x^2 = (4 + 3i)^2 = 4^2 + 2(4)(3i) + (3i)^2$$
$$x^2 = 16 + 24i + 9i^2$$
Since $i^2 = -1$,
$$x^2 = 16 + 24i - 9 = 7 + 24i$$
Step 2: Calculate $x^3$
$$x^3 = x \cdot x^2 = (4 + 3i)(7 + 24i)$$
$$x^3 = 4(7) + 4(24i) + 3i(7) + 3i(24i)$$
$$x^3 = 28 + 96i + 21i + 72i^2$$
$$x^3 = 28 + 117i - 72$$
$$x^3 = -44 + 117i$$
Step 3: Calculate $4x^2$
$$4x^2 = 4(7 + 24i)$$
$$4x^2 = 28 + 96i$$
Step 4: Calculate $7x$
$$7x = 7(4 + 3i)$$
$$7x = 28 + 21i$$
Step 5: Substitute the calculated values into the polynomial expression
The expression is $x^3 - 4x^2 - 7x + 12$.
Substitute the values from Steps 2, 3, and 4:
$$x^3 - 4x^2 - 7x + 12 = (-44 + 117i) - (28 + 96i) - (28 + 21i) + 12$$
Group the real and imaginary parts:
$$= (-44 - 28 - 28 + 12) + (117 - 96 - 21)i$$
Calculate the real part:
$$-44 - 28 - 28 + 12 = -72 - 28 + 12 = -100 + 12 = -88$$
Calculate the imaginary part:
$$117 - 96 - 21 = 21 - 21 = 0$$
Thus, the expression evaluates to:
$$= -88 + 0i = -88$$
Correct Answer: D