Expand and state your answer as a polynomial in standard form.
left parenthesis, 2, x, minus, y, to the power 5 , right parenthesis, squared
(2x−y
5
)
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
1 answer
To expand the expression \((2x - y^5)^2\) as a polynomial in standard form, we will follow the same process outlined in the example.
Step 1: Expand the Power
We need to expand the square of the binomial \((2x - y^5)\):
\[
(2x - y^5)(2x - y^5)
\]
Step 2: Apply the Distributive Property
We'll apply the distributive property (also known as the FOIL method for binomials) to expand:
\[
= (2x)(2x) + (2x)(-y^5) + (-y^5)(2x) + (-y^5)(-y^5)
\]
Step 3: Calculate Each Term
Now we calculate each term:
\((2x)(2x) = 4x^2\)
\((2x)(-y^5) = -2xy^5\)
\((-y^5)(2x) = -2xy^5\)
\((-y^5)(-y^5) = y^{10}\)
Step 4: Combine Like Terms
Now, we combine the like terms:
\[
4x^2 - 2xy^5 - 2xy^5 + y^{10}
\]
This simplifies to:
\[
4x^2 - 4xy^5 + y^{10}
\]
Step 5: Write in Standard Form
Finally, we present the polynomial in standard form (from highest degree to lowest):
\[
y^{10} - 4xy^5 + 4x^2
\]
Final Answer
Thus, the expansion of \((2x - y^5)^2\) in standard form is:
\[
y^{10} - 4xy^5 + 4x^2
\]
