Expand and state your answer as a polynomial in standard form.

Question

left parenthesis, 4, x, to the power 5 , minus, y, squared, right parenthesis, squared
(4x
5
−y
2
)
2

Solve the problem just like the example below
⬇️
Expand and state your answer as a polynomial in standard form.
left parenthesis, 3, x, to the power 5 , plus, y, to the power 5 , right parenthesis, squared
(3x
5
+y
5
)
2
For binomials left parenthesis adding or subtracting right parenthesis
For binomials (adding or subtracting)
we must expand the power.
we must expand the power.
left parenthesis, 3, x, to the power 5 , plus, y, to the power 5 , right parenthesis, left parenthesis, 3, x, to the power 5 , plus, y, to the power 5 , right parenthesis
(3x
5
+y
5
)(3x
5
+y
5
)
left parenthesis, 3, x, to the power 5 , right parenthesis, left parenthesis, 3, x, to the power 5 , right parenthesis, plus, left parenthesis, 3, x, to the power 5 , right parenthesis, left parenthesis, y, to the power 5 , right parenthesis, plus, left parenthesis, 3, x, to the power 5 , right parenthesis, left parenthesis, y, to the power 5 , right parenthesis, plus, left parenthesis, y, to the power 5 , right parenthesis, left parenthesis, y, to the power 5 , right parenthesis
(3x
5
)(3x
5
)+(3x
5
)(y
5
)+(3x
5
)(y
5
)+(y
5
)(y
5
)
9, x, to the power 10 , plus, 3, x, to the power 5 , y, to the power 5 , plus, 3, x, to the power 5 , y, to the power 5 , plus, y, to the power 10
9x
10
+3x
5
y
5
+3x
5
y
5
+y
10
9, x, to the power 10 , plus, 6, x, to the power 5 , y, to the power 5 , plus, y, to the power 10
9x
10
+6x
5
y
5
+y
10

Answer 1

To expand and express the polynomial \((4x^5 - y^2)^2\) in standard form, we can follow these steps:

Step 1: Set Up the Expression

We need to expand the expression \((4x^5 - y^2)(4x^5 - y^2)\).

Step 2: Apply the Distributive Property (or FOIL)

Now we will apply the distributive property:

\[ (4x^5 - y^2)(4x^5 - y^2) = (4x^5)(4x^5) + (4x^5)(-y^2) + (-y^2)(4x^5) + (-y^2)(-y^2) \]

Step 3: Calculate Each Term

1. First Term: \[ (4x^5)(4x^5) = 16x^{10} \]

2. Outer Term: \[ (4x^5)(-y^2) = -4x^5y^2 \]

3. Inner Term: \[ (-y^2)(4x^5) = -4x^5y^2 \]

4. Last Term: \[ (-y^2)(-y^2) = y^4 \]

Step 4: Combine Like Terms

Now, we combine like terms:

The two middle terms \(-4x^5y^2\) and \(-4x^5y^2\) can be combined: \[ -4x^5y^2 - 4x^5y^2 = -8x^5y^2 \]

So, putting it all together, we have: \[ 16x^{10} - 8x^5y^2 + y^4 \]

Step 5: Write in Standard Form

The standard form of the polynomial is: \[ \boxed{16x^{10} - 8x^5y^2 + y^4} \]