Question
\( 10 p^{3}+-11 p^{7} q+6 q+4 \) Simpliy \( \left(\frac{2}{3} x^{3}-\frac{1}{y^{2}}\right)\left(\frac{1}{2} x^{2}-\frac{4}{3} x y-\frac{1}{2} y^{2}\right)(y \neq 0 \)
Ask by Rogers Pierce. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified forms are:
1. \( -11 p^{7} q + 10 p^{3} + 6 q + 4 \)
2. \( \frac{1}{3}x^5 - \frac{8}{9}x^4 y - \frac{1}{3}x^3 y^{2} - \frac{1}{2}\frac{x^2}{y^2} + \frac{4}{3}\frac{x}{y} + \frac{1}{2} \)
Solution
Let's simplify each of the given expressions step by step.
---
### 1. Simplifying \( 10 p^{3} - 11 p^{7} q + 6 q + 4 \)
The expression \(10 p^{3} - 11 p^{7} q + 6 q + 4\) is a polynomial with distinct terms. Since none of the terms are like terms (they have different variables or exponents), the expression is already in its simplest form.
**Simplified Form:**
\[
-11 p^{7} q + 10 p^{3} + 6 q + 4
\]
---
### 2. Simplifying \( \left(\frac{2}{3} x^{3} - \frac{1}{y^{2}}\right) \left(\frac{1}{2} x^{2} - \frac{4}{3} x y - \frac{1}{2} y^{2}\right) \) (where \( y \neq 0 \))
To simplify the product of these two polynomials, we'll use the distributive property (also known as the FOIL method for binomials), multiplying each term in the first polynomial by each term in the second polynomial.
**Step-by-Step Multiplication:**
1. **Multiply \(\frac{2}{3} x^3\) by each term in the second polynomial:**
\[
\frac{2}{3} x^3 \times \frac{1}{2} x^2 = \frac{1}{3} x^5
\]
\[
\frac{2}{3} x^3 \times \left(-\frac{4}{3} x y\right) = -\frac{8}{9} x^4 y
\]
\[
\frac{2}{3} x^3 \times \left(-\frac{1}{2} y^2\right) = -\frac{1}{3} x^3 y^2
\]
2. **Multiply \(-\frac{1}{y^2}\) by each term in the second polynomial:**
\[
-\frac{1}{y^2} \times \frac{1}{2} x^2 = -\frac{1}{2} \frac{x^2}{y^2}
\]
\[
-\frac{1}{y^2} \times \left(-\frac{4}{3} x y\right) = \frac{4}{3} \frac{x}{y}
\]
\[
-\frac{1}{y^2} \times \left(-\frac{1}{2} y^2\right) = \frac{1}{2}
\]
**Combine All the Terms:**
\[
\frac{1}{3} x^5 - \frac{8}{9} x^4 y - \frac{1}{3} x^3 y^2 - \frac{1}{2} \frac{x^2}{y^2} + \frac{4}{3} \frac{x}{y} + \frac{1}{2}
\]
**Final Simplified Form:**
\[
\frac{1}{3}x^5 - \frac{8}{9}x^4 y - \frac{1}{3}x^3 y^{2} - \frac{1}{2}\frac{x^2}{y^2} + \frac{4}{3}\frac{x}{y} + \frac{1}{2}
\]
---
**Summary:**
1. **First Expression:**
\[
-11 p^{7} q + 10 p^{3} + 6 q + 4
\]
2. **Second Expression:**
\[
\frac{1}{3}x^5 - \frac{8}{9}x^4 y - \frac{1}{3}x^3 y^{2} - \frac{1}{2}\frac{x^2}{y^2} + \frac{4}{3}\frac{x}{y} + \frac{1}{2}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
Let's simplify the polynomial \( 10 p^{3} - 11 p^{7} q + 6 q + 4 \). We can rearrange it into standard form: \( -11 p^{7} q + 10 p^{3} + 6 q + 4 \). Since the terms can't be combined, this is the simplest expression in this form. For the product \( \left(\frac{2}{3} x^{3}-\frac{1}{y^{2}}\right)\left(\frac{1}{2} x^{2}-\frac{4}{3} x y-\frac{1}{2} y^{2}\right) \), we can use the distributive property (FOIL method) to multiply both polynomials. This will result in multiple terms, which are then simplified by combining like terms. The result is a cubic polynomial in \( x \) and \( y \) successfully expressed in one equation.
