The term containing the highest power of in th
polynomial is . Two of the roots of the
equation are -2 and 4 . Given that
is a quadratic factor of , find an
expression for in descending powers of .

Ask by Todd Harrington. in Singapore

Mar 24,2025

Question

The term containing the highest power of in th
polynomial is . Two of the roots of the
equation are -2 and 4 . Given that
is a quadratic factor of , find an
expression for in descending powers of .

Ask by Todd Harrington. in Singapore

Mar 24,2025

UpStudy AI · Accepted Answer

To find the expression for the polynomial $$ f(x) $$ in descending powers of $$ x $$, we can follow these steps:

1. **Identify the known factors**: We know that $$ f(x) $$ has roots at $$ x = -2 $$ and $$ x = 4 $$. This means that $$ (x + 2) $$ and $$ (x - 4) $$ are factors of $$ f(x) $$.

2. **Use the quadratic factor**: We are given that $$ x^2 - 7x + 2 $$ is a quadratic factor of $$ f(x) $$.

3. **Combine the factors**: Since $$ f(x) $$ is a polynomial, we can express it as:
   $$
   f(x) = k \cdot (x + 2)(x - 4)(x^2 - 7x + 2)
   $$
   where $$ k $$ is a constant.

4. **Determine the constant $$ k $$**: The leading term of $$ f(x) $$ is given as $$ 3x^4 $$. We will expand the product and find $$ k $$.

Let's first expand $$ (x + 2)(x - 4) $$:
$$
(x + 2)(x - 4) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8
$$

Now, we will multiply this result by the quadratic factor $$ (x^2 - 7x + 2) $$:
$$
f(x) = k \cdot (x^2 - 2x - 8)(x^2 - 7x + 2)
$$

Next, we will expand this product.

Let's denote:
- $$ A = x^2 - 2x - 8 $$
- $$ B = x^2 - 7x + 2 $$

Now, we will expand $$ A \cdot B $$:
$$
A \cdot B = (x^2 - 2x - 8)(x^2 - 7x + 2)
$$

We will perform the multiplication step by step.

1. Multiply $$ x^2 $$ by each term in $$ B $$:
   $$
   x^2 \cdot (x^2 - 7x + 2) = x^4 - 7x^3 + 2x^2
   $$

2. Multiply $$ -2x $$ by each term in $$ B $$:
   $$
   -2x \cdot (x^2 - 7x + 2) = -2x^3 + 14x^2 - 4x
   $$

3. Multiply $$ -8 $$ by each term in $$ B $$:
   $$
   -8 \cdot (x^2 - 7x + 2) = -8x^2 + 56x - 16
   $$

Now, combine all these results:
$$
f(x) = k \cdot (x^4 - 7x^3 + 2x^2 - 2x^3 + 14x^2 - 4x - 8x^2 + 56x - 16)
$$

Combine like terms:
$$
f(x) = k \cdot (x^4 - 9x^3 + 8x^2 + 52x - 16)
$$

Now, we need to find $$ k $$ such that the leading coefficient is 3. The leading term is $$ k \cdot 1 $$ (from $$ x^4 $$), so we set:
$$
k = 3
$$

Thus, the polynomial $$ f(x) $$ is:
$$
f(x) = 3(x^4 - 9x^3 + 8x^2 + 52x - 16)
$$

Now, let's expand this to get the final expression:
$$
f(x) = 3x^4 - 27x^3 + 24x^2 + 156x - 48
$$

So, the expression for $$ f(x) $$ in descending powers of $$ x $$ is:
$$
f(x) = 3x^4 - 27x^3 + 24x^2 + 156x - 48
$$
$$ f(x) = 3x^4 - 27x^3 + 24x^2 + 156x - 48 $$

The term containing the highest power of in th
polynomial is . Two of the roots of the
equation are -2 and 4 . Given that
is a quadratic factor of , find an
expression for in descending powers of .

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The term containing the highest power of in th polynomial is . Two of the roots of the equation are -2 and 4 . Given that is a quadratic factor of , find an expression for in descending powers of .

Solución de inteligencia artificial de Upstudy

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Latest Algebra Questions

The term containing the highest power of in th
polynomial is . Two of the roots of the
equation are -2 and 4 . Given that
is a quadratic factor of , find an
expression for in descending powers of .