Questions: Arrange in ascending powers of y. 10. 2y^2 + y^4 - 3y^3 + 9 11. 3x^2y - 5xy^4 - 2x^3 + 6y^2

Arrange in ascending powers of y. 10. 2y^2 + y^4 - 3y^3 + 9 11. 3x^2y - 5xy^4 - 2x^3 + 6y^2
Transcript text: Arrange in ascending powers of $y$. 10. $2 y^{2}+y^{4}-3 y^{3}+9$ 11. $3 x^{2} y-5 x y^{4}-2 x^{3}+6 y^{2}$
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Solution

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Solution Steps

To arrange the given expressions in ascending powers of \( y \), we need to sort the terms based on the exponent of \( y \) in each term.

For the first expression \( 2y^2 + y^4 - 3y^3 + 9 \):
  1. Identify the power of \( y \) in each term.
  2. Sort the terms based on these powers in ascending order.
For the second expression \( 3x^2y - 5xy^4 - 2x^3 + 6y^2 \):
  1. Identify the power of \( y \) in each term.
  2. Sort the terms based on these powers in ascending order.
Step 1: Identify the terms in the polynomial

For each polynomial, identify the individual terms and their powers of \( y \).

Polynomial 10:

\[ 2y^2 + y^4 - 3y^3 + 9 \]

Polynomial 11:

\[ 3x^2 y - 5xy^4 - 2x^3 + 6y^2 \]

Step 2: Arrange terms in ascending powers of \( y \)
Polynomial 10:

List the terms with their respective powers of \( y \):

  • \( 9 \) (constant term, \( y^0 \))
  • \( 2y^2 \) (\( y^2 \))
  • \( -3y^3 \) (\( y^3 \))
  • \( y^4 \) (\( y^4 \))

Arrange these terms in ascending order of \( y \): \[ 9 + 2y^2 - 3y^3 + y^4 \]

Polynomial 11:

List the terms with their respective powers of \( y \):

  • \( -2x^3 \) (no \( y \), \( y^0 \))
  • \( 3x^2 y \) (\( y^1 \))
  • \( 6y^2 \) (\( y^2 \))
  • \( -5xy^4 \) (\( y^4 \))

Arrange these terms in ascending order of \( y \): \[ -2x^3 + 3x^2 y + 6y^2 - 5xy^4 \]

Final Answer

Polynomial 10:

\[ \boxed{9 + 2y^2 - 3y^3 + y^4} \]

Polynomial 11:

\[ \boxed{-2x^3 + 3x^2 y + 6y^2 - 5xy^4} \]

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