Q. (x^{7} – 21x^{4} + 35x^{2} – 6x + 18).

Problem

Analyze and factor the polynomial:

\(x^7 – 21x^4 + 35x^2 – 6x + 18\)

Step 1: Identify the expression

The expression is:

\(x^7 – 21x^4 + 35x^2 – 6x + 18\)

This is a polynomial in one variable, \(x\).

The highest power of \(x\) is \(7\), so this is a seventh-degree polynomial.

Because the expression is not written as an equation, there is no single value to “solve for” unless we set the polynomial equal to zero.

So we will treat the task as factoring or simplifying the polynomial as much as possible.

Step 2: Check whether there are like terms to combine

The polynomial is:

\(x^7 – 21x^4 + 35x^2 – 6x + 18\)

Each term has a different power of \(x\):

\(x^7\)

\(-21x^4\)

\(35x^2\)

\(-6x\)

\(18\)

Since none of the variable terms have the same power of \(x\), there are no like terms to combine.

So the expression is already simplified in standard polynomial form.

Step 3: Check for a common factor

A common factor is something that divides every term in the expression.

The terms are:

\(x^7\), \(-21x^4\), \(35x^2\), \(-6x\), and \(18\)

The first four terms contain \(x\), but the constant term \(18\) does not.

So \(x\) is not a common factor of all terms.

Now check the numerical coefficients:

\(1\), \(-21\), \(35\), \(-6\), and \(18\)

The greatest common numerical factor of these coefficients is \(1\).

Therefore, there is no common factor other than \(1\).

Step 4: Try the Rational Root Theorem

To factor a polynomial over the rational numbers, we often check for rational roots.

If a polynomial has a rational root, then it has a linear factor of the form:

\(x – r\)

where \(r\) is the rational root.

The Rational Root Theorem says that possible rational roots come from the factors of the constant term divided by the factors of the leading coefficient.

The constant term is:

\(18\)

The leading coefficient is:

1

So the possible rational roots are the positive and negative factors of \(18\):

\(\pm 1,\ \pm 2,\ \pm 3,\ \pm 6,\ \pm 9,\ \pm 18\)

Step 5: Test possible rational roots

Let:

\(P(x) = x^7 – 21x^4 + 35x^2 – 6x + 18\)

Now test the possible rational roots.

Test \(x = 1\)

\(P(1) = 1^7 – 21(1^4) + 35(1^2) – 6(1) + 18\)

\(P(1) = 1 – 21 + 35 – 6 + 18\)

\(P(1) = 27\)

Since \(P(1) \ne 0\), \(x = 1\) is not a root.

Test \(x = -1\)

\(P(-1) = (-1)^7 – 21(-1)^4 + 35(-1)^2 – 6(-1) + 18\)

\(P(-1) = -1 – 21 + 35 + 6 + 18\)

\(P(-1) = 37\)

Since \(P(-1) \ne 0\), \(x = -1\) is not a root.

Test \(x = 2\)

\(P(2) = 2^7 – 21(2^4) + 35(2^2) – 6(2) + 18\)

\(P(2) = 128 – 21(16) + 35(4) – 12 + 18\)

\(P(2) = 128 – 336 + 140 – 12 + 18\)

\(P(2) = -62\)

Since \(P(2) \ne 0\), \(x = 2\) is not a root.

Test \(x = -2\)

\(P(-2) = (-2)^7 – 21(-2)^4 + 35(-2)^2 – 6(-2) + 18\)

\(P(-2) = -128 – 21(16) + 35(4) + 12 + 18\)

\(P(-2) = -128 – 336 + 140 + 12 + 18\)

\(P(-2) = -294\)

Since \(P(-2) \ne 0\), \(x = -2\) is not a root.

Test \(x = 3\)

\(P(3) = 3^7 – 21(3^4) + 35(3^2) – 6(3) + 18\)

\(P(3) = 2187 – 21(81) + 35(9) – 18 + 18\)

\(P(3) = 2187 – 1701 + 315\)

\(P(3) = 801\)

Since \(P(3) \ne 0\), \(x = 3\) is not a root.

Test \(x = -3\)

\(P(-3) = (-3)^7 – 21(-3)^4 + 35(-3)^2 – 6(-3) + 18\)

\(P(-3) = -2187 – 21(81) + 35(9) + 18 + 18\)

\(P(-3) = -2187 – 1701 + 315 + 18 + 18\)

\(P(-3) = -3537\)

Since \(P(-3) \ne 0\), \(x = -3\) is not a root.

Test \(x = 6\)

\(P(6) = 6^7 – 21(6^4) + 35(6^2) – 6(6) + 18\)

\(P(6) = 279936 – 21(1296) + 35(36) – 36 + 18\)

\(P(6) = 279936 – 27216 + 1260 – 36 + 18\)

\(P(6) = 253962\)

Since \(P(6) \ne 0\), \(x = 6\) is not a root.

Test \(x = -6\)

\(P(-6) = (-6)^7 – 21(-6)^4 + 35(-6)^2 – 6(-6) + 18\)

\(P(-6) = -279936 – 21(1296) + 35(36) + 36 + 18\)

\(P(-6) = -279936 – 27216 + 1260 + 36 + 18\)

\(P(-6) = -305838\)

Since \(P(-6) \ne 0\), \(x = -6\) is not a root.

Step 6: Decide whether the polynomial factors nicely over the rational numbers

The possible rational roots are:

\(\pm 1,\ \pm 2,\ \pm 3,\ \pm 6,\ \pm 9,\ \pm 18\)

Testing these values does not give \(0\).

That means the polynomial has no rational linear factor.

So the polynomial does not factor nicely into a product with a simple rational linear factor.

Step 7: Write the final simplified form

The polynomial is already simplified:

\(x^7 – 21x^4 + 35x^2 – 6x + 18\)

There are no like terms to combine.

There is no common factor to remove.

There is no rational root, so there is no simple rational linear factor.

Final answer

The expression is already simplified:

\(x^7 – 21x^4 + 35x^2 – 6x + 18\)

It has no common factor other than \(1\).

It has no rational roots from the Rational Root Theorem candidates:

\(\pm 1,\ \pm 2,\ \pm 3,\ \pm 6,\ \pm 9,\ \pm 18\)

Therefore, over the rational numbers, it does not have a simple linear factor.