- Question 5 simplifies to \(2y^8 - 4y^5 + 3y^4\).
- Question 6 results in \(5x^4 + 2x^3 + 4x^2 + 17x + 6\).
- Question 7 results in \(d^3 - 1\).
- Question 8 results in \(49x^2 + 84x + 36\).
Final answers: \(\boxed{2y^8 - 4y^5 + 3y^4, 5x^4 + 2x^3 + 4x^2 + 17x + 6, d^3 - 1, 49x^2 + 84x + 36}\)
Explanation
- Solving the algebraic expressions
The problem requires us to solve four separate algebraic expressions. Let's address them one by one.
Question 5: Simplify \(\frac{24 y^{12}-48 y^9+36 y^8}{12 y^4}\)
To simplify this expression, we divide each term in the numerator by the denominator:
\[ \frac{24 y^{12}}{12 y^4} - \frac{48 y^9}{12 y^4} + \frac{36 y^8}{12 y^4} \]
Using the exponent division rule \(\frac{y^m}{y^n} = y^{m-n}\), we get:
\[ \left(\frac{24}{12}\right)y^{12-4} - \left(\frac{48}{12}\right)y^{9-4} + \left(\frac{36}{12}\right)y^{8-4} = 2y^8 - 4y^5 + 3y^4 \]
Question 6: Subtract the polynomials \((5 x^4 + 3 x^3 + 11 x^2 + 10 x) - (x^3 + 7 x^2 - 7 x - 6)\)
Distribute the negative sign to each term in the second polynomial:
\[ 5 x^4 + 3 x^3 + 11 x^2 + 10 x - x^3 - 7 x^2 + 7 x + 6 \]
Now, group and combine the like terms (terms with the same exponent of \(x\)):
\[ 5 x^4 + (3 x^3 - x^3) + (11 x^2 - 7 x^2) + (10 x + 7 x) + 6 = 5x^4 + 2x^3 + 4x^2 + 17x + 6 \]
Question 7: Find the product \((d - 1)(d^2 + d + 1)\)
This is a standard difference of cubes pattern: \((a-b)(a^2+ab+b^2) = a^3-b^3\). Here, \(a=d\) and \(b=1\).
Applying the identity:
\[ (d - 1)(d^2 + d(1) + 1^2) = d^3 - 1^3 = d^3 - 1 \]
Question 8: Expand \((7 x + 6)^2\)
Using the binomial square formula \((a+b)^2 = a^2 + 2ab + b^2\) with \(a=7x\) and \(b=6\):
\[ (7x)^2 + 2(7x)(6) + 6^2 \]
\[ 49x^2 + 84x + 36 \]
Examples
Algebraic simplification is essential in real-world modeling. For instance, if you are designing a storage box and determine the volume as a polynomial in terms of its height \(x\), you might need to expand or simplify that expression to find the dimensions that optimize space. Calculating these expressions precisely ensures your design calculations are correct and efficient.