Factoring of polynomials

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1. To find the value of "c" such that the quadratic polynomial "(x^2) + 6x + c" is a perfect square, we note that "(x^2) + 6x + c" would have to be the square of some linear factor ax + b.

Thus

(x^2) + 6x + c = (ax + b)^2 = (ax + b)*(ax + b) = (a^2)*(x^2) + 2abx + (b^2)

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Comparing coefficients of x^2 (1 is the coefficient of x^2 in (x^2) + 6x + c, and a^2 is the coefficient of x^2 in (a^2)*(x^2) + 2abx + (b^2)), we get

1 = a^2

Thus either a = 1 or a = -1.

Since the coefficient of x^2 in "(x^2) + 6x + c" is positive, we can take "a" to be 1.

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Comparing coefficients of x^1 (6 is the coefficient of x^1 in (x^2) + 6x + c, and 2ab is the coefficient of x^1 in (a^2)*(x^2) + 2abx + (b^2)), we obtain

6 = 2ab

We already know that a = 1.

Substituting 1 for "a", we find that 6 = 2*1*b, which simplifies to 6 = 2b. Solving for "b", we get b = 6/2 = 3.

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Comparing the constant terms ("c" is the constant term in (x^2) + 6x + c, and b^2 is the constant term in (a^2)*(x^2) + 2abx + (b^2)), we find that

c = b^2

We already know that b = 3.

Substituting 3 for "b", we have c = 3^2 = 9.

Thus c = 9.

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2. To find the value of "a" such that the quadratic polynomial "(z^2) + 12z + a" is a perfect square, we note that "z^2 + 12z + a" would have to be the square of some linear factor bz + c.

Thus

(z^2) + 12z + a = (bz + c)^2 = (bz + c)*(bz + c) = (b^2)*(z*2) + 2bcz + (c^2)

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Comparing coefficients of z^2 (1 is the coefficient of z^2 in (z^2) + 12z + a, and b^2 is the coefficient of z^2 in (b^2)*(z*2) + 2bcz + (c^2)), we get

1 = b^2

Since the coefficient of z^2 in "(z^2) + 12z + a" is positive, we can take b to be 1.

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Comparing coefficients of z^1 (12 is the coefficient of z^1 in (z^2) + 12z + a, and 2bc is the coefficient of z^1 in (b^2)*(z*2) + 2bcz + (c^2)), we obtain

12 = 2bc

We already know that b = 1.

Substituting 1 for b, we obtain 12 = 2*1*c, which simplifies to 12 = 2c. Solving for "c", we get c = 12/2 = 6.

Thus c = 6.

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Comparing the constant terms ("a" is the constant term in (z^2) + 12z + a, and c^2 is the constant term in (b^2)*(z*2) + 2bcz + (c^2)), we find that

a = c^2

We already know that c = 6.

Substituting 6 for c, we see that a = 6^2 = 36.

Thus a = 36.

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3. To factor 3*(u^4)*(y^4) - 48*(u^4), first find the greatest common factor.

Note that 3 is a factor in both terms, and 3 is the constant factor of largest absolute value which is common to the two terms, so the constant factor in the greatest common factor is 3.

Also, u^4 is common to both terms, so u^4 is a factor in the greatest common factor.

The first term contains a factor of y^4, but the second term has no factor with y (raised to some power), so the greatest common factor is 3*(u^4).

Factoring out the greatest common factor, we find that

3*(u^4)*(y^4) - 48*(u^4) = ...