Answer :
Let's simplify each expression step-by-step.
### (viii) [tex]\(\left(2 x^3 + 7 x + 10 x^2\right) \times \frac{5}{2} x\)[/tex]
Expanding this expression:
[tex]\[ \left(2 x^3 + 7 x + 10 x^2\right) \times \frac{5}{2} x \][/tex]
First, distribute [tex]\(\frac{5}{2} x\)[/tex]:
[tex]\[ 2 x^3 \times \frac{5}{2} x + 7 x \times \frac{5}{2} x + 10 x^2 \times \frac{5}{2} x \][/tex]
Simplifying each term:
[tex]\[ (2 \times \frac{5}{2}) x^4 + (7 \times \frac{5}{2}) x^2 + (10 \times \frac{5}{2}) x^3 \][/tex]
[tex]\[ 5 x^4 + \frac{35}{2} x^2 + 25 x^3 \][/tex]
Rewriting it to a simpler polynomial format:
[tex]\[ x^2 \left(5 x^2 + 25 x + 17.5\right) \][/tex]
### (i) [tex]\(\left(8 x^3 - 4 x^2 + 5\right) \times \frac{1}{2} x + \left(3 x^2 - 5 x + 1\right) \times 4 x\)[/tex]
Expanding this expression:
[tex]\[ \left(8 x^3 - 4 x^2 + 5\right) \times \frac{1}{2} x + \left(3 x^2 - 5 x + 1\right) \times 4 x \][/tex]
Distributing [tex]\(\frac{1}{2} x\)[/tex] and [tex]\(4 x\)[/tex] respectively:
[tex]\[ \left(8 x^3 \times \frac{1}{2} x\right) + \left(-4 x^2 \times \frac{1}{2} x\right) + \left(5 \times \frac{1}{2} x\right) + \left(3 x^2 \times 4 x\right) + \left(-5 x \times 4 x\right) + \left(1 \times 4 x\right) \][/tex]
Simplifying each term:
[tex]\[ 4 x^4 - 2 x^3 + \frac{5}{2} x + 12 x^3 - 20 x^2 + 4 x \][/tex]
Combining like terms:
[tex]\[ 4 x^4 + 10 x^3 - 20 x^2 + 6.5 x \][/tex]
Rewriting it to a simpler polynomial format:
[tex]\[ x \left(4 x^3 + 10 x^2 - 20 x + 6.5\right) \][/tex]
### (ii) [tex]\(\left(5 a^3 - 18 a^2 - 25 a\right) \times \frac{1}{5} a - \left(9 a^2 - 6 a + 11\right) \times \frac{-2}{3} a\)[/tex]
Expanding this expression:
[tex]\[ \left(5 a^3 - 18 a^2 - 25 a\right) \times \frac{1}{5} a - \left(9 a^2 - 6 a + 11\right) \times \frac{-2}{3} a \][/tex]
Distributing [tex]\(\frac{1}{5} a\)[/tex] and [tex]\(\frac{-2}{3} a\)[/tex]:
[tex]\[ \left(5 a^3 \times \frac{1}{5} a\right) + \left(-18 a^2 \times \frac{1}{5} a\right) + \left(-25 a \times \frac{1}{5} a\right) - \left(9 a^2 \times \frac{-2}{3} a\right) + \left(-6 a \times \frac{-2}{3} a\right) + \left(11 \times \frac{-2}{3} a\right) \][/tex]
Simplifying each term:
[tex]\[ a^4 - \frac{18}{5} a^3 - 5 a^2 + \frac{18}{3} a^3 + 4 a^2 - \frac{22}{3} a \][/tex]
Combining like terms:
[tex]\[ a^4 + \left(\frac{54}{15} - \frac{18}{5}\right)a^3 + \left(-5 + 4 \right) a^2 - \frac{22}{3} a \][/tex]
[tex]\[ a^4 + \frac{2.4}{1} a^3 - 1 a^2 - \frac{7.3333}{1} a \][/tex]
Rewriting it to a simpler polynomial format:
[tex]\[ a \left(a^3 + 2.4 a^2 - 9 a + 7.3333\right) \][/tex]
### (iii) [tex]\(\left(x^2 + x y + \frac{1}{2} x y^2\right) \times 2 y + \left(y^2 + 2 x y - x^2\right) \times \frac{3}{2} x\)[/tex]
Expanding this expression:
[tex]\[ \left(x^2 + x y + \frac{1}{2} x y^2\right) \times 2 y + \left(y^2 + 2 x y - x^2\right) \times \frac{3}{2} x \][/tex]
Distributing [tex]\(2 y\)[/tex] and [tex]\(\frac{3}{2} x\)[/tex]:
[tex]\[ 2 y x^2 + 2 y x y + y^2 - \frac{3}{2} x^3 + y \times \frac{3}{2 x}\][/tex]
Simplifying each term:
[tex]\[ 2 x^2 y + 2 x y^2 + x y^3 + 0 x + \frac{3 x^2} \][/tex]
Combining like terms:
[tex]\[ 0 - \frac{x^3}{1} + 5x y^2 + y^3 \][/tex]
Rewriting it to a simpler polynomial format:
[tex]\[ x(-3x^2 + 5xy + y^3 + y^2\][/tex]
\]
### (iv) [tex]\(\left(9 x^2 - 12 x y + 4 y^2\right) \times x y^2 - \left(4 x^2 - 12 y^2 + 9 x y\right) \times x^2 y\)[/tex]
Expanding this expression:
[tex]\[ \left(9 x^2 - 12 x y + 4 y^2\right) \times x y^2 - \left(4 x^2 - 12 y^2 + 9 x y\right) \times x^2 y \][/tex]
Distributing [tex]\(x y^2\)[/tex] and [tex]\(x^2 y\)[/tex]:
[tex]\[ \left(9 x^2 x y^2\right) - \left(-12 x y x y^2\right) - \left(4 y^2 x y^2\right) + \left(4 x^2 + 12 x^2 \right) + \left{- 3x^2 y \right) \][/tex]
Simplifying each term:
Combining each term and Identifying similar
Combining like terms:
\[4 x y (-x^3 + y^3)
Rewriting it to a simpler polynomial format:
\(4xy - x^3 - y^3
So, all the simplifications are:
(viii) x (5 x^2 + 25 x + 17.5)
(i) x (4 x^3 + 10 x^2 + x - 6.5)
(ii)a^4 + \frac{2.4}{1} a^3 - 1 a^2 - \frac{7.3333}{1} a
(iii)
0 - \frac{x^3}{1} + x y^3 + y^
(iv)
\(y^2 (-x + y)
}
### (viii) [tex]\(\left(2 x^3 + 7 x + 10 x^2\right) \times \frac{5}{2} x\)[/tex]
Expanding this expression:
[tex]\[ \left(2 x^3 + 7 x + 10 x^2\right) \times \frac{5}{2} x \][/tex]
First, distribute [tex]\(\frac{5}{2} x\)[/tex]:
[tex]\[ 2 x^3 \times \frac{5}{2} x + 7 x \times \frac{5}{2} x + 10 x^2 \times \frac{5}{2} x \][/tex]
Simplifying each term:
[tex]\[ (2 \times \frac{5}{2}) x^4 + (7 \times \frac{5}{2}) x^2 + (10 \times \frac{5}{2}) x^3 \][/tex]
[tex]\[ 5 x^4 + \frac{35}{2} x^2 + 25 x^3 \][/tex]
Rewriting it to a simpler polynomial format:
[tex]\[ x^2 \left(5 x^2 + 25 x + 17.5\right) \][/tex]
### (i) [tex]\(\left(8 x^3 - 4 x^2 + 5\right) \times \frac{1}{2} x + \left(3 x^2 - 5 x + 1\right) \times 4 x\)[/tex]
Expanding this expression:
[tex]\[ \left(8 x^3 - 4 x^2 + 5\right) \times \frac{1}{2} x + \left(3 x^2 - 5 x + 1\right) \times 4 x \][/tex]
Distributing [tex]\(\frac{1}{2} x\)[/tex] and [tex]\(4 x\)[/tex] respectively:
[tex]\[ \left(8 x^3 \times \frac{1}{2} x\right) + \left(-4 x^2 \times \frac{1}{2} x\right) + \left(5 \times \frac{1}{2} x\right) + \left(3 x^2 \times 4 x\right) + \left(-5 x \times 4 x\right) + \left(1 \times 4 x\right) \][/tex]
Simplifying each term:
[tex]\[ 4 x^4 - 2 x^3 + \frac{5}{2} x + 12 x^3 - 20 x^2 + 4 x \][/tex]
Combining like terms:
[tex]\[ 4 x^4 + 10 x^3 - 20 x^2 + 6.5 x \][/tex]
Rewriting it to a simpler polynomial format:
[tex]\[ x \left(4 x^3 + 10 x^2 - 20 x + 6.5\right) \][/tex]
### (ii) [tex]\(\left(5 a^3 - 18 a^2 - 25 a\right) \times \frac{1}{5} a - \left(9 a^2 - 6 a + 11\right) \times \frac{-2}{3} a\)[/tex]
Expanding this expression:
[tex]\[ \left(5 a^3 - 18 a^2 - 25 a\right) \times \frac{1}{5} a - \left(9 a^2 - 6 a + 11\right) \times \frac{-2}{3} a \][/tex]
Distributing [tex]\(\frac{1}{5} a\)[/tex] and [tex]\(\frac{-2}{3} a\)[/tex]:
[tex]\[ \left(5 a^3 \times \frac{1}{5} a\right) + \left(-18 a^2 \times \frac{1}{5} a\right) + \left(-25 a \times \frac{1}{5} a\right) - \left(9 a^2 \times \frac{-2}{3} a\right) + \left(-6 a \times \frac{-2}{3} a\right) + \left(11 \times \frac{-2}{3} a\right) \][/tex]
Simplifying each term:
[tex]\[ a^4 - \frac{18}{5} a^3 - 5 a^2 + \frac{18}{3} a^3 + 4 a^2 - \frac{22}{3} a \][/tex]
Combining like terms:
[tex]\[ a^4 + \left(\frac{54}{15} - \frac{18}{5}\right)a^3 + \left(-5 + 4 \right) a^2 - \frac{22}{3} a \][/tex]
[tex]\[ a^4 + \frac{2.4}{1} a^3 - 1 a^2 - \frac{7.3333}{1} a \][/tex]
Rewriting it to a simpler polynomial format:
[tex]\[ a \left(a^3 + 2.4 a^2 - 9 a + 7.3333\right) \][/tex]
### (iii) [tex]\(\left(x^2 + x y + \frac{1}{2} x y^2\right) \times 2 y + \left(y^2 + 2 x y - x^2\right) \times \frac{3}{2} x\)[/tex]
Expanding this expression:
[tex]\[ \left(x^2 + x y + \frac{1}{2} x y^2\right) \times 2 y + \left(y^2 + 2 x y - x^2\right) \times \frac{3}{2} x \][/tex]
Distributing [tex]\(2 y\)[/tex] and [tex]\(\frac{3}{2} x\)[/tex]:
[tex]\[ 2 y x^2 + 2 y x y + y^2 - \frac{3}{2} x^3 + y \times \frac{3}{2 x}\][/tex]
Simplifying each term:
[tex]\[ 2 x^2 y + 2 x y^2 + x y^3 + 0 x + \frac{3 x^2} \][/tex]
Combining like terms:
[tex]\[ 0 - \frac{x^3}{1} + 5x y^2 + y^3 \][/tex]
Rewriting it to a simpler polynomial format:
[tex]\[ x(-3x^2 + 5xy + y^3 + y^2\][/tex]
\]
### (iv) [tex]\(\left(9 x^2 - 12 x y + 4 y^2\right) \times x y^2 - \left(4 x^2 - 12 y^2 + 9 x y\right) \times x^2 y\)[/tex]
Expanding this expression:
[tex]\[ \left(9 x^2 - 12 x y + 4 y^2\right) \times x y^2 - \left(4 x^2 - 12 y^2 + 9 x y\right) \times x^2 y \][/tex]
Distributing [tex]\(x y^2\)[/tex] and [tex]\(x^2 y\)[/tex]:
[tex]\[ \left(9 x^2 x y^2\right) - \left(-12 x y x y^2\right) - \left(4 y^2 x y^2\right) + \left(4 x^2 + 12 x^2 \right) + \left{- 3x^2 y \right) \][/tex]
Simplifying each term:
Combining each term and Identifying similar
Combining like terms:
\[4 x y (-x^3 + y^3)
Rewriting it to a simpler polynomial format:
\(4xy - x^3 - y^3
So, all the simplifications are:
(viii) x (5 x^2 + 25 x + 17.5)
(i) x (4 x^3 + 10 x^2 + x - 6.5)
(ii)a^4 + \frac{2.4}{1} a^3 - 1 a^2 - \frac{7.3333}{1} a
(iii)
0 - \frac{x^3}{1} + x y^3 + y^
(iv)
\(y^2 (-x + y)
}
