Math  /  Algebra

QuestionExpand and state your answer as a polynomial in standard form. (4x5+y5)2\left(4 x^{5}+y^{5}\right)^{2}

Studdy Solution

STEP 1

1. We are asked to expand the expression (4x5+y5)2(4x^5 + y^5)^2.
2. The expression is a binomial squared, which can be expanded using the binomial theorem or by applying the distributive property.

STEP 2

1. Apply the formula for squaring a binomial.
2. Simplify the resulting expression.
3. Write the polynomial in standard form.

STEP 3

Apply the formula for squaring a binomial:
(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
In this case, a=4x5a = 4x^5 and b=y5b = y^5.

STEP 4

Substitute a=4x5a = 4x^5 and b=y5b = y^5 into the formula:
(4x5+y5)2=(4x5)2+2(4x5)(y5)+(y5)2(4x^5 + y^5)^2 = (4x^5)^2 + 2(4x^5)(y^5) + (y^5)^2

STEP 5

Calculate each term:
1. (4x5)2=16x10(4x^5)^2 = 16x^{10}
2. 2(4x5)(y5)=8x5y52(4x^5)(y^5) = 8x^5y^5
3. (y5)2=y10(y^5)^2 = y^{10}

STEP 6

Combine the terms to form the expanded polynomial:
16x10+8x5y5+y1016x^{10} + 8x^5y^5 + y^{10}

STEP 7

Write the polynomial in standard form, which is already achieved:
16x10+8x5y5+y1016x^{10} + 8x^5y^5 + y^{10}

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