What Is the Fully Factored Form of 32a3 12a2?

The fully factored form of 32a3 12a2 is 2a(16a2 6a). This can be seen by expanding the parentheses and distributing the 2a term.

In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 12 can be factored into 6 × 2, where 6 and 2 are the factors of 12. In general, a polynomial with coefficients in some given field or ring can be factored into a product of polynomials of degree one with coefficients in the same field or ring.

The aim of factoring polynomials is usually to reduce them to a form that is easier to work with, or to extract factors that may have some special properties. For example, the quadratic polynomial x2 − 5x + 6 can be factored as (x − 3)(x − 2), and since the factors x − 3 and x − 2 are both linear polynomials, this gives us some information about the nature of the solutions to the equation x2 − 5x + 6 = 0. In fact, since the factors are linear, we know that the solutions must be real numbers, and since the product of the factors is 6, we also know that the solutions must be 2 and 3.

There are a number of different methods that can be used to factor polynomials, and the choice of method will usually depend on the type of polynomial that is to be factorized. For example, if the polynomial is a quadratic, then the quadratic formula can be used to find the roots, and from these the factors can be determined. If the polynomial is a cubic, then one approach is to use the method of substitution, by substituting x = y − b/3a to transform the cubic into a quadratic in y, which can then be factorized using the quadratic formula.

Another approach to factorizing polynomials is to use the Factor Theorem. This theorem states that if a polynomial f(x) has a root r, then x − r is a factor of f(x). For example, if we want to factor the polynomial x2 + 5x + 6, then we can use the Factor Theorem to test whether x − 2 is a factor. We know that 2 is a root of the polyn

The factors of 32a3 12a2 are the numbers that divide evenly into 32a3 12a2. The factors of 32a3 12a2 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, and 144.

In mathematics, a prime polynomial is a polynomial that has no non-zero factors other than itself and 1. So, is 32a3 + 12a2 a prime polynomial?

It is easy to see that 32a3 + 12a2 is not a prime number, because it can be factorised as 4(8a3 + 3a2). However, it is not so easy to see whether 32a3 + 12a2 is a prime polynomial.

To factorise a polynomial, we need to find its factors. Factors are numbers or polynomials that, when multiplied together, give us the original polynomial. For example, the factors of 32a3 + 12a2 are 4(8a3 + 3a2).

To find the factors of 32a3 + 12a2, we could use trial and error, but that would be very time-consuming. A quicker method is to use the Factor Theorem.

The Factor Theorem tells us that if a polynomial can be divided by (x - a) then a is a factor of the polynomial. So, to find the factors of 32a3 + 12a2, we need to find values of x that make the expression (x - a) divide into 32a3 + 12a2.

It is not difficult to find two values of x that make (x - a) divide into 32a3 + 12a2. For example, if we take x = 4, then (4 - a) = 4 - 4a will divide into 32a3 + 12a2. We can also take x = -2, then (-2 - a) = -2 - (-2a) will also divide into 32a3 + 12a2.

So, the factors of 32a3 + 12a2 are (x - 4) and (x - (-2)). We can factorise 32a3 + 12a2 as follows:

32a3 + 12a2 = (x - 4)(x - (-2))

Therefore, 32a3 + 12a2 is not a prime polynomial.

There is no one definitive answer to this question. The greatest common factor of 32a3 12a2 could be any number that evenly divides both 32a3 and 12a2. Some possible candidates for the greatest common factor of 32a3 12a2 include 2, 4, a, and 12a2. However, it is also possible that the greatest common factor of 32a3 12a2 is something else entirely. In short, the greatest common factor of 32a3 12a2 is dependent on the specific values of a and 2.

There is no definitive answer to this question as the answer is dependent on the interpretation of the question. It could be argued that the degree of 32a3 12a2 is 12, as this is the highest exponent of the a term, or it could be argued that the degree of 32a3 12a2 is 9, as this is the highest overall exponent when both terms are taken into account. Ultimately, it is up to the reader to decide what they believe the degree of 32a3 12a2 to be.

In mathematics, the leading coefficient of a polynomial is the coefficient of the term of the highest degree. For example, the leading coefficient of 32a3 12a2 is 32.

The leading coefficient is often denoted by the letter a (for "arrow"), lc (for "leading coefficient"), or simply the leading term. In the example above, the leading term is 32a3.

The leading coefficient can be used to determine the behavior of a polynomial at infinity. For example, a polynomial with a leading coefficient of 1 will approach infinity asymptotically (i.e., will get arbitrarily large in absolute value, but not change sign), while a polynomial with a leading coefficient of -1 will approach negative infinity.

More generally, the leading coefficient can be used to bound the growth of a polynomial. For example, if the leading coefficient is 1, then the polynomial will grow no faster than the exponential function; if the leading coefficient is -1, then the polynomial will grow no slower than the exponential function.

In the example above, the leading coefficient is 32. This means that the polynomial will grow no faster than 32x3.

In mathematics, a constant term is a term in an algebraic expression that has a fixed value and is not the result of a multiplication or division. The constant term of 32a3 12a2 is 12a2. This is because the 3 and the 2 in the expression are not the result of a multiplication or division, and so they have a fixed value. The 12 in the expression is also a constant term. This is because it is not the result of a multiplication or division, and so it has a fixed value as well.

In mathematics, an imaginary root of a polynomial is a complex number that satisfies the equation. For example, the imaginary roots of the equation x^2+1=0 are the complex numbers square root of negative one, denoted byi.

The equation 32a^3+12a^2=0 has three solutions, two of which are imaginary roots. In order to find these solutions, we can use the quadratic equation, which states that if a polynomial of degree two has the form:

Ax^2+Bx+C=0

Then its solutions are:

x=(-B+-sqrt(B^2-4AC))/(2A)

In our equation, we have A=32, B=12, and C=0. Plugging these values into the quadratic equation, we get:

x=(-12+-sqrt(144-0))/(2(32))

x=(-12+-sqrt(144))/(64)

x=(-12+-12i)/64

x1=(-12+12i)/64

x2=(-12-12i)/64

We can see that x1 and x2 are our imaginary roots, as they are complex numbers that satisfy the equation.