Objective: Find the coefficient of the term containing $x$ in the expansion of $(x + 9)(x - 6)(x + 5)$.
Finding the Coefficient of x
The expansion of a product of three binomials, $(a + b)(c + d)(e + f)$, results in a cubic polynomial. The term containing $x$ can be obtained by multiplying the $x$ term from exactly one binomial with the constant terms from the other two.
For the expression $(x + 9)(x - 6)(x + 5)$, we identify three combinations that yield an $x$ term:
- Multiply the $x$ from $(x + 9)$, the constant $-6$ from $(x - 6)$, and the constant $5$ from $(x + 5)$.
Calculation:x \times (-6) \times 5 = -30x$ - Multiply the constant $9$ from $(x + 9)$, the $x$ from $(x - 6)$, and the constant $5$ from $(x + 5)$.
Calculation:9 \times x \times 5 = 45x$ - Multiply the constant $9$ from $(x + 9)$, the constant $-6$ from $(x - 6)$, and the $x$ from $(x + 5)$.
Calculation:9 \times (-6) \times x = -54x$
Summing the x Terms
To find the total coefficient of $x$ in the expanded form, sum the coefficients from these three terms:
Total coefficient = (-30) + 45 + (-54)$
Total coefficient = 15 - 54$
Total coefficient = -39$
Therefore, the coefficient of $x$ in the expansion is -39.
Important Questions from Algebra (Notes)
- If $(y-12) = 4\sqrt{5}$, then find the value of $\sqrt{y-3} - \frac{1}{\sqrt{y-3}}$.
- If $x^2 + \frac{1}{x^2} = 16$ and $x \neq 0$, then what is the value of $x^4 + \frac{1}{x^4}$?
- Find the value of $\frac{x+3}{x^2-2x} \times \frac{2x-1}{x^2+2x+4} \times \frac{x^4-8x}{2x^2+5x-3}$
- The roots of the equation $ax^3-24x^2+188x-480=0$ are three consecutive even natural numbers. The value of a is _____.
- If $a = 0.1125$, then find the value of $100\left[\sqrt{1 + 2(3a) + 9a^2} - 4a\right]$.
