$$ x^{3}+5 x^{2}-4 x-20 \text { yo } 8 $$ 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?

Question

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 $$ q = 49 $$.

For what value of and the expression
will be the perfect square?

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Algebra Questions

For what value of and the expression will be the perfect square?