A circular mirror is surrounded by a square metal frame. The radius of the mirror is 7x. The side length of the metal frame is 21x. What is the area of the metal frame?
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Part 1
The area of the frame is
enter your response here square units.
(Type your answer in factored form. Type an exact answer in terms of pi.)
The area of the frame can be calculated by subtracting the area of the mirror from the area of the frame.
The area of the frame can be found by finding the area of the square and subtracting the area of the circle.
The area of the square is (side length)^2, so the area of the square frame is (21x)^2 = 441x^2.
The area of the circle is π(radius)^2, so the area of the mirror is π(7x)^2 = 49πx^2.
Therefore, the area of the frame is 441x^2 - 49πx^2 = (441 - 49π)x^2.
So, the area of the metal frame is (441 - 49π)x^2 square units.
Factor the polynomial.
17 x Superscript 4 Baseline y cubed plus 51 x Superscript 6 Baseline y
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Part 1
17x Superscript 4y cubedplus51x Superscript 6yequals
enter your response here (Factor completely.)
To factor the polynomial 17x^4y^3 + 51x^6y, we can look for common factors in each term.
The common factor here is 17x^4y, which we can factor out:
17x^4y^3 + 51x^6y = 17x^4y(x^2y^2 + 3x^2)
Now, let's look at the expression in the parentheses (x^2y^2 + 3x^2). We can factor out another common factor of x^2:
17x^4y(x^2y^2 + 3x^2) = 17x^4y(x^2(y^2 + 3))
So the completely factored form of the polynomial 17x^4y^3 + 51x^6y is 17x^4y(x^2)(y^2 + 3).