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Math  /  Algebra

QuestionQuestion
Expand and state your answer as a polynomial in standard form. (2x2−y)2\left(2 x^{2}-y\right)^{2}

Studdy Solution

STEP 1

1. We are given an expression (2x2−y)2(2x^2 - y)^2 that needs to be expanded.
2. The expansion will involve using the binomial theorem or the distributive property.
3. The final result should be expressed as a polynomial in standard form.

STEP 2

1. Apply the binomial expansion formula to (2x2−y)2(2x^2 - y)^2.
2. Simplify the resulting expression.
3. Write the polynomial in standard form.

STEP 3

Apply the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 to the expression (2x2−y)2(2x^2 - y)^2.
Let a=2x2a = 2x^2 and b=yb = y.
(2x2−y)2=(2x2)2−2(2x2)(y)+y2 (2x^2 - y)^2 = (2x^2)^2 - 2(2x^2)(y) + y^2

STEP 4

Calculate each term in the expansion:
1. Calculate (2x2)2(2x^2)^2:
(2x2)2=4x4 (2x^2)^2 = 4x^4
2. Calculate −2(2x2)(y)-2(2x^2)(y):
−2(2x2)(y)=−4x2y -2(2x^2)(y) = -4x^2y
3. Calculate y2y^2:
y2=y2 y^2 = y^2

STEP 5

Combine all the terms from the expansion:
4x4−4x2y+y2 4x^4 - 4x^2y + y^2

STEP 6

Write the polynomial in standard form, which is already achieved:
4x4−4x2y+y2 4x^4 - 4x^2y + y^2
The expanded polynomial in standard form is:
4x4−4x2y+y2 \boxed{4x^4 - 4x^2y + y^2}

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