Polynomial $x^4$ Coefficient Calculation
We need to determine the coefficient of the $x^4$ term in the expansion of the polynomial $(x - 1)^3(x – 2)^3$.
Step 1: Expand $(x - 1)^3$
Using the binomial expansion $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$: $(x - 1)^3 = x^3 - 3(x^2)(1) + 3(x)(1)^2 - 1^3$ $(x - 1)^3 = x^3 - 3x^2 + 3x - 1$
Step 2: Expand $(x - 2)^3$
Using the same binomial expansion formula: $(x - 2)^3 = x^3 - 3(x^2)(2) + 3(x)(2)^2 - 2^3$ $(x - 2)^3 = x^3 - 6x^2 + 12x - 8$
Step 3: Identify $x^4$ terms in the product
Let $P(x) = x^3 - 3x^2 + 3x - 1$ and $Q(x) = x^3 - 6x^2 + 12x - 8$. We multiply $P(x)$ by $Q(x)$. To find the $x^4$ term, we identify pairs of terms, one from each polynomial, whose powers sum to 4:
- From $P(x)$'s $x^3$ term and $Q(x)$'s $x^1$ term: $(x^3) \times (12x) = 12x^4$. The coefficient is $1 \times 12 = 12$.
- From $P(x)$'s $x^2$ term and $Q(x)$'s $x^2$ term: $(-3x^2) \times (-6x^2) = 18x^4$. The coefficient is $(-3) \times (-6) = 18$.
- From $P(x)$'s $x^1$ term and $Q(x)$'s $x^3$ term: $(3x) \times (x^3) = 3x^4$. The coefficient is $3 \times 1 = 3$.
Step 4: Sum the coefficients
The total coefficient for the $x^4$ term is the sum of the coefficients found in Step 3: $12 + 18 + 3 = 33$
Final Answer
The coefficient of $x^4$ in the polynomial $(x - 1)^3(x – 2)^3$ is 33.
Important Questions from Algebra
- For positive non-zero real variables $x$ and $y$, if
$ln\left(\frac{x+y}{2}\right) = \frac{1}{2} [ln\left(x\right) + ln\left(y\right)]$
then, the value of $\frac{x}{y} + \frac{y}{x}$ is Given $f(x, y) = x^2 - 2xy + y^2$
The complete contour of the equation $f(x, y) = 1$ is described by the option(s) ___.
- It is given that $x$ and $y$ are integers in the following equation:
$$(x + y - 7)^2 + (y + 3x - 13)^2 = 0$$
The value of $(x^3 + y^3)$ is ________ (in integer). - If $pqr \neq 0$ and $p^{-x} = \frac{1}{q}$, $q^{-y} = \frac{1}{r}$, $r^{-z} = \frac{1}{p}$, what is the value of the product $xyz$?
- Two points $(4, p)$ and $(0, q)$ lie on a straight line having a slope of $3/4$. The value of $(p – q)$ is
