Simplify (x^5-3x^3+px^3+q)/(x^2+x-2)
The question provided is asking for the simplification of a rational expression—a polynomial divided by another polynomial. In this case, the polynomial in the numerator is x^5 - 3x^3 + px^3 + q, and the one in the denominator is x^2 + x - 2. The problem likely involves factoring both the numerator and the denominator to possibly cancel out common factors, performing polynomial long division, or possibly synthetic division if applicable, to simplify the expression to its lowest terms. The variables p and q represent coefficients that are to be possibly determined or are simply part of the expression to be simplified.
$\frac{x^{5} - 3 x^{3} + p x^{3} + q}{x^{2} + x - 2}$
Answer
Solution:
Step 1:
Identify two numbers whose multiplication equals $c$ and addition equals $b$ in the quadratic expression $x^2 + bx + c$. For our problem, we need two numbers that multiply to $-2$ and add up to $1$. The numbers are $-1$ and $2$.
Step 2:
Factor the denominator using the numbers found in Step 1 to get $\frac{x^5 - 3x^3 + px^3 + q}{(x - 1)(x + 2)}$.
Knowledge Notes:
To simplify a rational expression where the numerator is a polynomial and the denominator is a quadratic expression, one approach is to factor the quadratic expression. Factoring is the process of breaking down an expression into a product of simpler expressions. The quadratic expression can be factored by finding two numbers that multiply to give the constant term $c$ and add to give the coefficient of the linear term $b$ in the quadratic expression $x^2 + bx + c$.
In this case, we are given the quadratic expression $x^2 + x - 2$. We need to find two numbers that multiply to $-2$ (the constant term) and add up to $1$ (the coefficient of the linear term). The numbers that satisfy these conditions are $-1$ and $2$ because $(-1) \cdot 2 = -2$ and $(-1) + 2 = 1$.
Once these numbers are found, we can write the quadratic expression as a product of two binomials: $(x - 1)(x + 2)$. This is known as factoring by grouping. The numerator does not change in this process, and we rewrite the entire expression with the factored denominator. The final step would be to simplify further if possible, by canceling out common factors in the numerator and denominator, but this is not shown in the given steps.
