$$ x^{3}+5 x^{2}-4 x-20 \text { yo } 8 $$ For what value of $$ p $$ and $$ q $$ the expression $$ x^{4}+8 x^{3}+30 x^{2}+p x+q $$ will be the perfect square?

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UpStudy AI · Accepted Answer

To determine the values of $$ p $$ and $$ q $$ such that the polynomial

$$
x^{4} + 8x^{3} + 30x^{2} + px + q
$$

is a perfect square, we can express it as the square of a quadratic polynomial:

$$
(x^{2} + ax + b)^{2} = x^{4} + 2ax^{3} + (a^{2} + 2b)x^{2} + 2abx + b^{2}
$$

By expanding and comparing coefficients with the given polynomial, we obtain the following system of equations:

$$
\begin{cases}
2a = 8 & \text{(Coefficient of } x^{3}) \
a^{2} + 2b = 30 & \text{(Coefficient of } x^{2}) \
2ab = p & \text{(Coefficient of } x) \
b^{2} = q & \text{(Constant term)}
\end{cases}
$$

**Step 1: Solve for $$ a $$**

$$
2a = 8 \implies a = 4
$$

**Step 2: Solve for $$ b $$**

$$
a^{2} + 2b = 30 \implies 16 + 2b = 30 \implies 2b = 14 \implies b = 7
$$

**Step 3: Determine $$ p $$ and $$ q $$**

$$
p = 2ab = 2 \times 4 \times 7 = 56
$$
$$
q = b^{2} = 7^{2} = 49
$$

**Conclusion:**

The values that make the polynomial a perfect square are:

$$
p = 56 \quad \text{and} \quad q = 49
$$

**Verification:**

Let's verify that the polynomial becomes a perfect square with these values:

$$
(x^{2} + 4x + 7)^{2} = x^{4} + 8x^{3} + 30x^{2} + 56x + 49
$$

This matches the original polynomial when $$ p = 56 $$ and $$ q = 49 $$.

**Final Answer:**

$$
p = 56 \quad \text{and} \quad q = 49
$$
$$ p = 56 $$ and $$ q = 49 $$.

\[ x^{3}+5 x^{2}-4 x-20 \text { yo } 8 \] For what value of \( p \) and \( q \) the expression \( x^{4}+8 x^{3}+30 x^{2}+p x+q \) will be the perfect square?

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