$\frac{13.3 a^2+16 a+7}{\left(3 a^{-}\right)\left(a^{+}\right)}$

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bri638212 bri638212

02-11-2026
Mathematics

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$\frac{13.3 a^2+16 a+7}{\left(3 a^{-}\right)\left(a^{+}\right)}$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T19:15:35-05:00

The numerator $13.3a^2 + 16a + 7$ has a negative discriminant ($-116.4$), indicating no real roots and an irreducible form over real numbers.
The denominator, expanded as $3a^2 + (3y - x)a - xy$, cannot be made proportional to the numerator using real numbers $x$ and $y$.
Since no common factors exist, the expression $\frac{13.3a^2 + 16a + 7}{(3a - x)(a + y)}$ cannot be simplified by cancellation.
The expression remains in its rational form $\boxed{\frac{13.3a^2 + 16a + 7}{(3a - x)(a + y)}}$ for any real $x, y$.

Explanation

Analyzing the Numerator

We are presented with a rational expression $\frac{13.3a^2 + 16a + 7}{(3a - \dots)(a + \dots)}$. To analyze this expression, we first examine the quadratic numerator $P(a) = 13.3a^2 + 16a + 7$. To determine if the expression can be simplified through factor cancellation, we check if this quadratic has real roots. The discriminant of $P(a) = 13.3a^2 + 16a + 7$ is given by $\Delta = b^2 - 4ac = 16^2 - 4(13.3)(7) = 256 - 372.4 = -116.4$. Since the discriminant is negative, the numerator has no real roots, meaning it cannot be factored into real linear factors. Consequently, the expression cannot be simplified by canceling a linear factor from the denominator.

Expanding and Analyzing the Denominator

To understand the structure of the denominator $(3a - x)(a + y)$, where $x$ and $y$ are the missing components represented by the ellipses, we expand the product: $(3a - x)(a + y) = 3a^2 + 3ay - ax - xy = 3a^2 + (3y - x)a - xy$. If the expression is to be analyzed as a simplified rational function or a ratio, we might aim for the ratio of the coefficients of the numerator to the denominator to be a constant $C$. Setting up the proportionality, we have $\frac{13.3}{3} = \frac{16}{3y - x} = \frac{7}{-xy} = C$. From the first part, $C = \frac{13.3}{3} \approx 4.433$.

Conclusion on Simplification

Using the constant $C = \frac{133}{30}$, we can find values for $x$ and $y$ such that the ratio holds. This leads to the system of equations: $3y - x = \frac{16}{C} = \frac{16 \times 30}{133} = \frac{480}{133} \approx 3.61$ and $-xy = \frac{7}{C} = \frac{7 \times 30}{133} = \frac{210}{133} \approx 1.58$. Substituting $x = 3y - \frac{480}{133}$ into $-xy = \frac{210}{133}$, we obtain a quadratic equation in $y$: $-(3y - \frac{480}{133})y = \frac{210}{133}$, which simplifies to $-3y^2 + \frac{480}{133}y = \frac{210}{133}$, or $3y^2 - \frac{480}{133}y + \frac{210}{133} = 0$. Solving this for $y$ provides values for the missing components. Given that the numerator has no real roots, any real-valued $x$ and $y$ will result in an irreducible rational expression unless the expression is a constant multiple of a squared form. Since the discriminant of this new quadratic in $y$ is $\Delta_y = (\frac{480}{133})^2 - 4(3)(\frac{210}{133}) \approx 13.0 - 18.9 < 0$, there are no real solutions for $x$ and $y$ that make the denominator proportional to the numerator. Thus, the expression remains irreducible.

Examples

This type of analysis is common in control theory or filter design in engineering, where one must match the poles (the roots of the denominator) and zeros (the roots of the numerator) of a transfer function to achieve a desired output. If the zeros and poles cannot be matched, it signifies that the system cannot be simplified into a simpler gain component, helping engineers understand the inherent complexity of the system they are modeling.