Answer :
Answer:
Step-by-step explanation:To make the polynomial \( y^4 + 6y^3 + 19y^2 + 30y \) a perfect square, we need to find a polynomial \( P(y) \) such that:
\[ y^4 + 6y^3 + 19y^2 + 30y + k = (y^2 + ay + b)^2 \]
where \( k \) is the constant we need to find.
First, let's expand \( (y^2 + ay + b)^2 \):
\[ (y^2 + ay + b)^2 = y^4 + 2ay^3 + (a^2 + 2b)y^2 + 2aby + b^2 \]
Now, we need to match coefficients from \( y^4 + 6y^3 + 19y^2 + 30y + k \) and \( y^4 + 2ay^3 + (a^2 + 2b)y^2 + 2aby + b^2 \):
1. Coefficient of \( y^4 \):
\[ 1 = 1 \]
(This is always true.)
2. Coefficient of \( y^3 \):
\[ 6 = 2a \]
\[ a = 3 \]
3. Coefficient of \( y^2 \):
\[ 19 = a^2 + 2b \]
\[ 19 = 3^2 + 2b \]
\[ 19 = 9 + 2b \]
\[ 2b = 10 \]
\[ b = 5 \]
4. Coefficient of \( y \):
\[ 30 = 2ab \]
\[ 30 = 2 \cdot 3 \cdot 5 \]
\[ 30 = 30 \]
(This is true.)
5. Constant term:
\[ k = b^2 \]
\[ k = 5^2 \]
\[ k = 25 \]
Thus, the constant \( k \) that must be added to \( y^4 + 6y^3 + 19y^2 + 30y \) to make it a perfect square is \( \boxed{25} \).
To find what must be added to \( y^4 + 6y^3 + 19y^2 + 30y \) to make it a perfect square, we will denote the term to be added by \( k \). So we want to find \( k \) such that:
\[ y^4 + 6y^3 + 19y^2 + 30y + k \]
is a perfect square.
Let's assume the perfect square is of the form \( (y^2 + ay + b)^2 \). Expanding this, we get:
\[ (y^2 + ay + b)^2 = y^4 + 2ay^3 + (a^2 + 2b)y^2 + 2aby + b^2 \]
We will match the coefficients from \( y^4 + 6y^3 + 19y^2 + 30y + k \) to the expanded form:
1. For \( y^4 \): Coefficient is 1.
2. For \( y^3 \): Coefficient is \( 2a \). So, \( 2a = 6 \Rightarrow a = 3 \).
3. For \( y^2 \): Coefficient is \( a^2 + 2b \). So, \( 3^2 + 2b = 19 \Rightarrow 9 + 2b = 19 \Rightarrow 2b = 10 \Rightarrow b = 5 \).
4. For \( y \): Coefficient is \( 2ab \). So, \( 2 \cdot 3 \cdot 5 = 30 \). This matches.
5. The constant term \( b^2 \) will give us \( k \).
Now, substituting \( a = 3 \) and \( b = 5 \) into the form:
\[ (y^2 + 3y + 5)^2 = y^4 + 6y^3 + 9y^2 + 10y^2 + 30y + 25 \]
\[ = y^4 + 6y^3 + 19y^2 + 30y + 25 \]
Thus, the term \( k \) that must be added is:
\[ k = 25 \]
So, 25 must be added to \( y^4 + 6y^3 + 19y^2 + 30y \) to make it a perfect square.